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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssceq 16201 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)

Theoremrescval 16202 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)       ((𝐶𝑉𝐻𝑊) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Theoremrescval2 16203 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   (𝜑𝐶𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))       (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Theoremrescbas 16204 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑆𝐵)       (𝜑𝑆 = (Base‘𝐷))

Theoremreschom 16205 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑆𝐵)       (𝜑𝐻 = (Hom ‘𝐷))

Theoremreschomf 16206 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑆𝐵)       (𝜑𝐻 = (Homf𝐷))

Theoremrescco 16207 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑆𝐵)    &    · = (comp‘𝐶)       (𝜑· = (comp‘𝐷))

Theoremrescabs 16208 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝐽 Fn (𝑇 × 𝑇))    &   (𝜑𝑆𝑊)    &   (𝜑𝑇𝑆)       (𝜑 → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))

Theoremrescabs2 16209 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐶𝑉)    &   (𝜑𝐽 Fn (𝑇 × 𝑇))    &   (𝜑𝑆𝑊)    &   (𝜑𝑇𝑆)       (𝜑 → ((𝐶s 𝑆) ↾cat 𝐽) = (𝐶cat 𝐽))

Theoremissubc 16210* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐻 = (Homf𝐶)    &    1 = (Id‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑆 = dom dom 𝐽)       (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))

Theoremissubc2 16211* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐻 = (Homf𝐶)    &    1 = (Id‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐽 Fn (𝑆 × 𝑆))       (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))

Theorem0ssc 16212 For any category 𝐶, the empty set is a subcategory subset of 𝐶. (Contributed by AV, 23-Apr-2020.)
(𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))

Theorem0subcat 16213 For any category 𝐶, the empty set is a (full) subcategory of 𝐶, see example 4.3(1.a) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)
(𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))

Theoremcatsubcat 16214 For any category 𝐶, 𝐶 itself is a (full) subcategory of 𝐶, see example 4.3(1.b) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)
(𝐶 ∈ Cat → (Homf𝐶) ∈ (Subcat‘𝐶))

Theoremsubcssc 16215 An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   𝐻 = (Homf𝐶)       (𝜑𝐽cat 𝐻)

Theoremsubcfn 16216 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 16217 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 16218 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 16219 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 16220 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 16221* 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 16222 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 16223 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐽 ∈ (Subcat‘𝐶))       (𝜑𝐷 ∈ Cat)

Theoremissubc3 16224* 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 17063, 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 16225 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 16226 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 16227 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 16228 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 16229 Extend class notation with the class of all functors.
class Func

Syntaxcidfu 16230 Extend class notation with identity functor.
class idfunc

Syntaxccofu 16231 Extend class notation with functor composition.
class func

Syntaxcresf 16232 Extend class notation to include restriction of a functor to a subcategory.
class f

Definitiondf-func 16233* 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 16234* 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 16235* 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 16236* Define the restriction of a functor to a subcategory (analogue of df-res 4944). (Contributed by Mario Carneiro, 6-Jan-2017.)
f = (𝑓 ∈ V, ∈ V ↦ ⟨((1st𝑓) ↾ dom dom ), (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥)))⟩)

Theoremrelfunc 16237 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Rel (𝐷 Func 𝐸)

Theoremfuncrcl 16238 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))

Theoremisfunc 16239* 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 16240* 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 16241 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 16242* 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 16243 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 16244 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 16245 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 16246 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 16247 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 16248 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 16249 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 16250 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 16251* 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 16252 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 16253 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 16254 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 16255 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 16256 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 16257* 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 16258 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 16259 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 16260 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 16261 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 16262* 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 16263 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 16264 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 16265 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 16266 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 16267* 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 16268* 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 16269 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 16270 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 16271 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 16272* 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 16273 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 16274 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 16275 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 16276 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 16277 Extend class notation with the class of all full functors.
class Full

Syntaxcfth 16278 Extend class notation with the class of all faithful functors.
class Faith

Definitiondf-full 16279* 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 16280* Function returning all the faithful 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 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 16281 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)

Theoremfthfunc 16282 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)

Theoremrelfull 16283 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Rel (𝐶 Full 𝐷)

Theoremrelfth 16284 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Rel (𝐶 Faith 𝐷)

Theoremisfull 16285* 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 16286* 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 16287 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 16288* 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 16289* 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 16290* 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 16291* 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 16292 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 16293 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 16294 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 16295 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 16296 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 16297 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 16298 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 16299 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 16300 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)))

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