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Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubcfn 17101 An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝑆 = dom dom 𝐽)       (𝜑𝐽 Fn (𝑆 × 𝑆))
 
Theoremsubcss1 17102 The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &   𝐵 = (Base‘𝐶)       (𝜑𝑆𝐵)
 
Theoremsubcss2 17103 The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))
 
Theoremsubcidcl 17104 The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &   (𝜑𝑋𝑆)    &    1 = (Id‘𝐶)       (𝜑 → ( 1𝑋) ∈ (𝑋𝐽𝑋))
 
Theoremsubccocl 17105 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &   (𝜑𝑋𝑆)    &    · = (comp‘𝐶)    &   (𝜑𝑌𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐹 ∈ (𝑋𝐽𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐽𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍))
 
Theoremsubccatid 17106* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &    1 = (Id‘𝐶)       (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑥𝑆 ↦ ( 1𝑥))))
 
Theoremsubcid 17107 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &    1 = (Id‘𝐶)    &   (𝜑𝑋𝑆)       (𝜑 → ( 1𝑋) = ((Id‘𝐷)‘𝑋))
 
Theoremsubccat 17108 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐽 ∈ (Subcat‘𝐶))       (𝜑𝐷 ∈ Cat)
 
Theoremissubc3 17109* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 17959, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐻 = (Homf𝐶)    &    1 = (Id‘𝐶)    &   𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐽 Fn (𝑆 × 𝑆))       (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))
 
Theoremfullsubc 17110 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory), see definition 4.1(2) of [Adamek] p. 48. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑆𝐵)       (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))
 
Theoremfullresc 17111 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑆𝐵)    &   𝐷 = (𝐶s 𝑆)    &   𝐸 = (𝐶cat (𝐻 ↾ (𝑆 × 𝑆)))       (𝜑 → ((Homf𝐷) = (Homf𝐸) ∧ (compf𝐷) = (compf𝐸)))
 
Theoremresscat 17112 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) ∈ Cat)
 
Theoremsubsubc 17113 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐷 = (𝐶cat 𝐻)       (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻)))
 
8.1.7  Functors
 
Syntaxcfunc 17114 Extend class notation with the class of all functors.
class Func
 
Syntaxcidfu 17115 Extend class notation with identity functor.
class idfunc
 
Syntaxccofu 17116 Extend class notation with functor composition.
class func
 
Syntaxcresf 17117 Extend class notation to include restriction of a functor to a subcategory.
class f
 
Definitiondf-func 17118* Function returning all the functors from a category 𝑡 to a category 𝑢. Definition 3.17 of [Adamek] p. 29, and definition in [Lang] p. 62 ("covariant functor"). Intuitively a functor associates any morphism of 𝑡 to a morphism of 𝑢, any object of 𝑡 to an object of 𝑢, and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of 𝑡 to an object of 𝑢 we write it associates any identity of 𝑡 to an identity of 𝑢 which simplifies the definition. According to remark 3.19 in [Adamek] p. 30, "a functor F : A -> B is technically a family of functions; one from Ob(A) to Ob(B) [here: f, called "the object part" in the following], and for each pair (A,A') of A-objects, one from hom(A,A') to hom(FA, FA') [here: g, called "the morphism part" in the following]". (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑢)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑢)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))})
 
Definitiondf-idfu 17119* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc = (𝑡 ∈ Cat ↦ (Base‘𝑡) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
 
Definitiondf-cofu 17120* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩)
 
Definitiondf-resf 17121* Define the restriction of a functor to a subcategory (analogue of df-res 5561). (Contributed by Mario Carneiro, 6-Jan-2017.)
f = (𝑓 ∈ V, ∈ V ↦ ⟨((1st𝑓) ↾ dom dom ), (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥)))⟩)
 
Theoremrelfunc 17122 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Rel (𝐷 Func 𝐸)
 
Theoremfuncrcl 17123 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
 
Theoremisfunc 17124* Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &    1 = (Id‘𝐷)    &   𝐼 = (Id‘𝐸)    &    · = (comp‘𝐷)    &   𝑂 = (comp‘𝐸)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)       (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵𝐶𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
 
Theoremisfuncd 17125* Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &    1 = (Id‘𝐷)    &   𝐼 = (Id‘𝐸)    &    · = (comp‘𝐷)    &   𝑂 = (comp‘𝐸)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)    &   (𝜑𝐹:𝐵𝐶)    &   (𝜑𝐺 Fn (𝐵 × 𝐵))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))    &   ((𝜑𝑥𝐵) → ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))       (𝜑𝐹(𝐷 Func 𝐸)𝐺)
 
Theoremfuncf1 17126 The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)       (𝜑𝐹:𝐵𝐶)
 
Theoremfuncixp 17127* The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)       (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
 
Theoremfuncf2 17128 The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))
 
Theoremfuncfn2 17129 The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐷)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)       (𝜑𝐺 Fn (𝐵 × 𝐵))
 
Theoremfuncid 17130 A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &    1 = (Id‘𝐷)    &   𝐼 = (Id‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)       (𝜑 → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋)))
 
Theoremfuncco 17131 A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐷)    &    · = (comp‘𝐷)    &   𝑂 = (comp‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑀 ∈ (𝑋𝐻𝑌))    &   (𝜑𝑁 ∈ (𝑌𝐻𝑍))       (𝜑 → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀)))
 
Theoremfuncsect 17132 The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝑆 = (Sect‘𝐷)    &   𝑇 = (Sect‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀(𝑋𝑆𝑌)𝑁)       (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
 
Theoremfuncinv 17133 The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐼 = (Inv‘𝐷)    &   𝐽 = (Inv‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀(𝑋𝐼𝑌)𝑁)       (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
 
Theoremfunciso 17134 The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐼 = (Iso‘𝐷)    &   𝐽 = (Iso‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀 ∈ (𝑋𝐼𝑌))       (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
 
Theoremfuncoppc 17135 A functor on categories yields a functor on the opposite categories (in the same direction), see definition 3.41 of [Adamek] p. 39. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑𝐹(𝐶 Func 𝐷)𝐺)       (𝜑𝐹(𝑂 Func 𝑃)tpos 𝐺)
 
Theoremidfuval 17136* Value of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
 
Theoremidfu2nd 17137 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
 
Theoremidfu2 17138 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝑋(2nd𝐼)𝑌)‘𝐹) = 𝐹)
 
Theoremidfu1st 17139 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑 → (1st𝐼) = ( I ↾ 𝐵))
 
Theoremidfu1 17140 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ((1st𝐼)‘𝑋) = 𝑋)
 
Theoremidfucl 17141 The identity functor is a functor. Example 3.20(1) of [Adamek] p. 30. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)       (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
 
Theoremcofuval 17142* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))       (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
 
Theoremcofu1st 17143 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))       (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))
 
Theoremcofu1 17144 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))    &   (𝜑𝑋𝐵)       (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
 
Theoremcofu2nd 17145 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
 
Theoremcofu2 17146 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝑋(2nd ‘(𝐺func 𝐹))𝑌)‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))
 
Theoremcofuval2 17147* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑𝐻(𝐷 Func 𝐸)𝐾)       (𝜑 → (⟨𝐻, 𝐾⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐻𝐹), (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
 
Theoremcofucl 17148 The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))       (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Func 𝐸))
 
Theoremcofuass 17149 Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐻 ∈ (𝐷 Func 𝐸))    &   (𝜑𝐾 ∈ (𝐸 Func 𝐹))       (𝜑 → ((𝐾func 𝐻) ∘func 𝐺) = (𝐾func (𝐻func 𝐺)))
 
Theoremcofulid 17150 The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   𝐼 = (idfunc𝐷)       (𝜑 → (𝐼func 𝐹) = 𝐹)
 
Theoremcofurid 17151 The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   𝐼 = (idfunc𝐶)       (𝜑 → (𝐹func 𝐼) = 𝐹)
 
Theoremresfval 17152* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐹𝑉)    &   (𝜑𝐻𝑊)       (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))⟩)
 
Theoremresfval2 17153* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐹𝑉)    &   (𝜑𝐻𝑊)    &   (𝜑𝐺𝑋)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))       (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
 
Theoremresf1st 17154 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 17155 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 17156 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 17157* 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 17158 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 17159 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 17160 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 17161 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 17162 Extend class notation with the class of all full functors.
class Full
 
Syntaxcfth 17163 Extend class notation with the class of all faithful functors.
class Faith
 
Definitiondf-full 17164* 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 17165* 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 17166 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
 
Theoremfthfunc 17167 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
 
Theoremrelfull 17168 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Rel (𝐶 Full 𝐷)
 
Theoremrelfth 17169 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Rel (𝐶 Faith 𝐷)
 
Theoremisfull 17170* 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 17171* 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 17172 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 17173* 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 17174* 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 17175* 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 17176* 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 17177 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 17178 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 17179 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 17180 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 17181 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 17182 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 17183 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 17184 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 17185 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 17186 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 17187 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 17188 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 17189 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 17190* 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 17191 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 17192 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 17193 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 17194 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 17195 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 17196 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 17197 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 17198 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 17199 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 17200 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))
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