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Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresf1st 16601 Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐹𝑉)    &   (𝜑𝐻𝑊)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))       (𝜑 → (1st ‘(𝐹f 𝐻)) = ((1st𝐹) ↾ 𝑆))
 
Theoremresf2nd 16602 Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐹𝑉)    &   (𝜑𝐻𝑊)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
 
Theoremfuncres 16603 A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐻 ∈ (Subcat‘𝐶))       (𝜑 → (𝐹f 𝐻) ∈ ((𝐶cat 𝐻) Func 𝐷))
 
Theoremfuncres2b 16604* Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑅 ∈ (Subcat‘𝐷))    &   (𝜑𝑅 Fn (𝑆 × 𝑆))    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))       (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺))
 
Theoremfuncres2 16605 A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝑅 ∈ (Subcat‘𝐷) → (𝐶 Func (𝐷cat 𝑅)) ⊆ (𝐶 Func 𝐷))
 
Theoremwunfunc 16606 A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐶𝑈)    &   (𝜑𝐷𝑈)       (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
 
Theoremfuncpropd 16607 If two categories have the same set of objects, morphisms, and compositions, then they have the same functors. (Contributed by Mario Carneiro, 17-Jan-2017.)
(𝜑 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
 
Theoremfuncres2c 16608 Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐸 = (𝐷s 𝑆)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝐴𝑆)       (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
 
8.1.8  Full & faithful functors
 
Syntaxcful 16609 Extend class notation with the class of all full functors.
class Full
 
Syntaxcfth 16610 Extend class notation with the class of all faithful functors.
class Faith
 
Definitiondf-full 16611* Function returning all the full functors from a category 𝐶 to a category 𝐷. A full functor is a functor in which all the morphism maps 𝐺(𝑋, 𝑌) between objects 𝑋, 𝑌𝐶 are surjections. Definition 3.27(3) in [Adamek] p. 34. (Contributed by Mario Carneiro, 26-Jan-2017.)
Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
 
Definitiondf-fth 16612* Function returning all the faithful functors from a category 𝐶 to a category 𝐷. A faithful functor is a functor in which all the morphism maps 𝐺(𝑋, 𝑌) between objects 𝑋, 𝑌𝐶 are injections. Definition 3.27(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 26-Jan-2017.)
Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
 
Theoremfullfunc 16613 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
 
Theoremfthfunc 16614 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
 
Theoremrelfull 16615 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Rel (𝐶 Full 𝐷)
 
Theoremrelfth 16616 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Rel (𝐶 Faith 𝐷)
 
Theoremisfull 16617* Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)       (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
 
Theoremisfull2 16618* Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐻 = (Hom ‘𝐶)       (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))))
 
Theoremfullfo 16619 The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Full 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌)))
 
Theoremfulli 16620* The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Full 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))       (𝜑 → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑅 = ((𝑋𝐺𝑌)‘𝑓))
 
Theoremisfth 16621* Value of the set of faithful functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)       (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦)))
 
Theoremisfth2 16622* Equivalent condition for a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)       (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))))
 
Theoremisffth2 16623* A fully faithful functor is a functor which is bijective on hom-sets. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)       (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦))))
 
Theoremfthf1 16624 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)))
 
Theoremfthi 16625 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))    &   (𝜑𝑆 ∈ (𝑋𝐻𝑌))       (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
 
Theoremffthf1o 16626 The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹𝑋)𝐽(𝐹𝑌)))
 
Theoremfullpropd 16627 If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))
 
Theoremfthpropd 16628 If two categories have the same set of objects, morphisms, and compositions, then they have the same faithful functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))
 
Theoremfulloppc 16629 The opposite functor of a full functor is also full. Proposition 3.43(d) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑𝐹(𝐶 Full 𝐷)𝐺)       (𝜑𝐹(𝑂 Full 𝑃)tpos 𝐺)
 
Theoremfthoppc 16630 The opposite functor of a faithful functor is also faithful. Proposition 3.43(c) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)       (𝜑𝐹(𝑂 Faith 𝑃)tpos 𝐺)
 
Theoremffthoppc 16631 The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺)       (𝜑𝐹((𝑂 Full 𝑃) ∩ (𝑂 Faith 𝑃))tpos 𝐺)
 
Theoremfthsect 16632 A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀 ∈ (𝑋𝐻𝑌))    &   (𝜑𝑁 ∈ (𝑌𝐻𝑋))    &   𝑆 = (Sect‘𝐶)    &   𝑇 = (Sect‘𝐷)       (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
 
Theoremfthinv 16633 A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀 ∈ (𝑋𝐻𝑌))    &   (𝜑𝑁 ∈ (𝑌𝐻𝑋))    &   𝐼 = (Inv‘𝐶)    &   𝐽 = (Inv‘𝐷)       (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
 
Theoremfthmon 16634 A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))    &   𝑀 = (Mono‘𝐶)    &   𝑁 = (Mono‘𝐷)    &   (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))       (𝜑𝑅 ∈ (𝑋𝑀𝑌))
 
Theoremfthepi 16635 A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))    &   𝐸 = (Epi‘𝐶)    &   𝑃 = (Epi‘𝐷)    &   (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑃(𝐹𝑌)))       (𝜑𝑅 ∈ (𝑋𝐸𝑌))
 
Theoremffthiso 16636 A fully faithful functor reflects isomorphisms. Corollary 3.32 of [Adamek] p. 35. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐹(𝐶 Full 𝐷)𝐺)    &   𝐼 = (Iso‘𝐶)    &   𝐽 = (Iso‘𝐷)       (𝜑 → (𝑅 ∈ (𝑋𝐼𝑌) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))))
 
Theoremfthres2b 16637* Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑅 ∈ (Subcat‘𝐷))    &   (𝜑𝑅 Fn (𝑆 × 𝑆))    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))       (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Faith (𝐷cat 𝑅))𝐺))
 
Theoremfthres2c 16638 Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐸 = (𝐷s 𝑆)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝐴𝑆)       (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Faith 𝐸)𝐺))
 
Theoremfthres2 16639 A faithful functor into a restricted category is also a faithful functor into the whole category. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝑅 ∈ (Subcat‘𝐷) → (𝐶 Faith (𝐷cat 𝑅)) ⊆ (𝐶 Faith 𝐷))
 
Theoremidffth 16640 The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐼 = (idfunc𝐶)       (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
 
Theoremcofull 16641 The composition of two full functors is full. Proposition 3.30(d) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Full 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Full 𝐸))       (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Full 𝐸))
 
Theoremcofth 16642 The composition of two faithful functors is faithful. Proposition 3.30(c) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Faith 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Faith 𝐸))       (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Faith 𝐸))
 
Theoremcoffth 16643 The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
(𝜑𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))    &   (𝜑𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))       (𝜑 → (𝐺func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
 
Theoremrescfth 16644 The inclusion functor from a subcategory is a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   𝐼 = (idfunc𝐷)       (𝐽 ∈ (Subcat‘𝐶) → 𝐼 ∈ (𝐷 Faith 𝐶))
 
Theoremressffth 16645 The inclusion functor from a full subcategory is a full and faithful functor, see also remark 4.4(2) in [Adamek] p. 49. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐷 = (𝐶s 𝑆)    &   𝐼 = (idfunc𝐷)       ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
 
Theoremfullres2c 16646 Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐸 = (𝐷s 𝑆)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝐴𝑆)       (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Full 𝐸)𝐺))
 
Theoremffthres2c 16647 Condition for a fully faithful functor to also be a fully faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐸 = (𝐷s 𝑆)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝐴𝑆)       (𝜑 → (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺𝐹((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))𝐺))
 
8.1.9  Natural transformations and the functor category
 
Syntaxcnat 16648 Extend class notation to include the collection of natural transformations.
class Nat
 
Syntaxcfuc 16649 Extend class notation to include the functor category.
class FuncCat
 
Definitiondf-nat 16650* Definition of a natural transformation between two functors. A natural transformation 𝐴:𝐹𝐺 is a collection of arrows 𝐴(𝑥):𝐹(𝑥)⟶𝐺(𝑥), such that 𝐴(𝑦) ∘ 𝐹() = 𝐺() ∘ 𝐴(𝑥) for each morphism :𝑥𝑦. Definition 6.1 in [Adamek] p. 83, and definition in [Lang] p. 65. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥))}))
 
Definitiondf-fuc 16651* Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. Definition 6.15 in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))))⟩})
 
Theoremfnfuc 16652 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
FuncCat Fn (Cat × Cat)
 
Theoremnatfval 16653* Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)       𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
 
Theoremisnat 16654* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)    &   (𝜑𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑𝐾(𝐶 Func 𝐷)𝐿)       (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))))
 
Theoremisnat2 16655* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))       (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴X𝑥𝐵 (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩ · ((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘)) = (((𝑥(2nd𝐺)𝑦)‘)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐺)‘𝑦))(𝐴𝑥)))))
 
Theoremnatffn 16656 The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)       𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))
 
Theoremnatrcl 16657 Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)       (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
 
Theoremnat1st2nd 16658 Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (𝐹𝑁𝐺))       (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
 
Theoremnatixp 16659* A natural transformation is a function from the objects of 𝐶 to homomorphisms from 𝐹(𝑥) to 𝐺(𝑥). (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))    &   𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)       (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
 
Theoremnatcl 16660 A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))    &   𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐴𝑋) ∈ ((𝐹𝑋)𝐽(𝐾𝑋)))
 
Theoremnatfn 16661 A natural transformation is a function on the objects of 𝐶. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))    &   𝐵 = (Base‘𝐶)       (𝜑𝐴 Fn 𝐵)
 
Theoremnati 16662 Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐷)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋)))
 
Theoremwunnat 16663 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐶𝑈)    &   (𝜑𝐷𝑈)       (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
 
Theoremcatstr 16664 A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.)
{⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} Struct ⟨1, 15⟩
 
Theoremfucval 16665* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝐵 = (𝐶 Func 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝐴 = (Base‘𝐶)    &    · = (comp‘𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑 = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))       (𝜑𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
 
Theoremfuccofval 16666* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝐵 = (𝐶 Func 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝐴 = (Base‘𝐶)    &    · = (comp‘𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &    = (comp‘𝑄)       (𝜑 = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
 
Theoremfucbas 16667 The objects of the functor category are functors from 𝐶 to 𝐷. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)       (𝐶 Func 𝐷) = (Base‘𝑄)
 
Theoremfuchom 16668 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)       𝑁 = (Hom ‘𝑄)
 
Theoremfucco 16669* Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝐴 = (Base‘𝐶)    &    · = (comp‘𝐷)    &    = (comp‘𝑄)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))    &   (𝜑𝑆 ∈ (𝐺𝑁𝐻))       (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))
 
Theoremfuccoval 16670 Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝐴 = (Base‘𝐶)    &    · = (comp‘𝐷)    &    = (comp‘𝑄)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))    &   (𝜑𝑆 ∈ (𝐺𝑁𝐻))    &   (𝜑𝑋𝐴)       (𝜑 → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑋) = ((𝑆𝑋)(⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩ · ((1st𝐻)‘𝑋))(𝑅𝑋)))
 
Theoremfuccocl 16671 The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &    = (comp‘𝑄)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))    &   (𝜑𝑆 ∈ (𝐺𝑁𝐻))       (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
 
Theoremfucidcl 16672 The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &    1 = (Id‘𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))       (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))
 
Theoremfuclid 16673 Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &    = (comp‘𝑄)    &    1 = (Id‘𝐷)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))       (𝜑 → (( 1 ∘ (1st𝐺))(⟨𝐹, 𝐺 𝐺)𝑅) = 𝑅)
 
Theoremfucrid 16674 Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &    = (comp‘𝑄)    &    1 = (Id‘𝐷)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))       (𝜑 → (𝑅(⟨𝐹, 𝐹 𝐺)( 1 ∘ (1st𝐹))) = 𝑅)
 
Theoremfucass 16675 Associativity of natural transformation composition. Remark 6.14(b) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &    = (comp‘𝑄)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))    &   (𝜑𝑆 ∈ (𝐺𝑁𝐻))    &   (𝜑𝑇 ∈ (𝐻𝑁𝐾))       (𝜑 → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)(⟨𝐹, 𝐺 𝐾)𝑅) = (𝑇(⟨𝐹, 𝐻 𝐾)(𝑆(⟨𝐹, 𝐺 𝐻)𝑅)))
 
Theoremfuccatid 16676* The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &    1 = (Id‘𝐷)       (𝜑 → (𝑄 ∈ Cat ∧ (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1st𝑓)))))
 
Theoremfuccat 16677 The functor category is a category. Remark 6.16 in [Adamek] p. 88. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑𝑄 ∈ Cat)
 
Theoremfucid 16678 The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝐼 = (Id‘𝑄)    &    1 = (Id‘𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))       (𝜑 → (𝐼𝐹) = ( 1 ∘ (1st𝐹)))
 
Theoremfucsect 16679* Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   𝑆 = (Sect‘𝑄)    &   𝑇 = (Sect‘𝐷)       (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥))))
 
Theoremfucinv 16680* Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   𝐼 = (Inv‘𝑄)    &   𝐽 = (Inv‘𝐷)       (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥))))
 
Theoreminvfuc 16681* If 𝑉(𝑥) is an inverse to 𝑈(𝑥) for each 𝑥, and 𝑈 is a natural transformation, then 𝑉 is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   𝐼 = (Inv‘𝑄)    &   𝐽 = (Inv‘𝐷)    &   (𝜑𝑈 ∈ (𝐹𝑁𝐺))    &   ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))𝑋)       (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))
 
Theoremfuciso 16682* A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   𝐼 = (Iso‘𝑄)    &   𝐽 = (Iso‘𝐷)       (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))))
 
Theoremnatpropd 16683 If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴 ∈ Cat)    &   (𝜑𝐵 ∈ Cat)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
 
Theoremfucpropd 16684 If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴 ∈ Cat)    &   (𝜑𝐵 ∈ Cat)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))
 
8.1.10  Initial, terminal and zero objects of a category
 
Syntaxcinito 16685 Extend class notation with the class of initial objects of a category.
class InitO
 
Syntaxctermo 16686 Extend class notation with the class of terminal objects of a category.
class TermO
 
Syntaxczeroo 16687 Extend class notation with the class of zero objects of a category.
class ZeroO
 
Definitiondf-inito 16688* An object A is said to be an initial object provided that for each object B there is exactly one morphism from A to B. Definition 7.1 in [Adamek] p. 101, or definition in [Lang] p. 57 (called "a universally repelling object" there). (Contributed by AV, 3-Apr-2020.)
InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)})
 
Definitiondf-termo 16689* An object A is called a terminal object provided that for each object B there is exactly one morphism from B to A. Definition 7.4 in [Adamek] p. 102, or definition in [Lang] p. 57 (called "a universally attracting object" there). (Contributed by AV, 3-Apr-2020.)
TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)})
 
Definitiondf-zeroo 16690 An object A is called a zero object provided that it is both an initial object and a terminal object. Definition 7.7 of [Adamek] p. 103. (Contributed by AV, 3-Apr-2020.)
ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
 
Theoreminitorcl 16691 Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020.)
(𝐼 ∈ (InitO‘𝐶) → 𝐶 ∈ Cat)
 
Theoremtermorcl 16692 Reverse closure for a terminal object: If a class has a terminal object, the class is a category. (Contributed by AV, 4-Apr-2020.)
(𝑇 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)
 
Theoremzeroorcl 16693 Reverse closure for a zero object: If a class has a zero object, the class is a category. (Contributed by AV, 4-Apr-2020.)
(𝑍 ∈ (ZeroO‘𝐶) → 𝐶 ∈ Cat)
 
Theoreminitoval 16694* The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
 
Theoremtermoval 16695* The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
 
Theoremzerooval 16696 The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
 
Theoremisinito 16697* The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐼𝐵)       (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
 
Theoremistermo 16698* The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐼𝐵)       (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
 
Theoremiszeroo 16699 The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐼𝐵)       (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶))))
 
Theoremisinitoi 16700* Implication of a class being an initial object. (Contributed by AV, 6-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)       ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
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