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Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
8.1.9  Natural transformations and the functor category
 
Syntaxcnat 17201 Extend class notation to include the collection of natural transformations.
class Nat
 
Syntaxcfuc 17202 Extend class notation to include the functor category.
class FuncCat
 
Definitiondf-nat 17203* 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 17204* 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 17205 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
FuncCat Fn (Cat × Cat)
 
Theoremnatfval 17206* Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
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 17207* 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 17208* 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 17209 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 17210 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 17211 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 17212* 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 17213 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 17214 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 17215 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 17216 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 17217 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 17218* 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 17219* 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 17220 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 17221 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 17222* 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 17223 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 17224 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 17225 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 17226 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 17227 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 17228 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 17229* 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 17230 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 17231 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 17232* 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 17233* 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 17234* 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 17235* 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 17236 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 17237 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 17238 Extend class notation with the class of initial objects of a category.
class InitO
 
Syntaxctermo 17239 Extend class notation with the class of terminal objects of a category.
class TermO
 
Syntaxczeroo 17240 Extend class notation with the class of zero objects of a category.
class ZeroO
 
Definitiondf-inito 17241* 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 17242* 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 17243 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 17244 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 17245 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 17246 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 17247* 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 17248* 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 17249 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 17250* 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 17251* 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 17252 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 17253* 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)))
 
Theoremistermoi 17254* Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)       ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
 
Theoreminitoid 17255 For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 6-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)       ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})
 
Theoremtermoid 17256 For a terminal object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 18-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)       ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})
 
Theoreminitoo 17257 An initial object is an object. (Contributed by AV, 14-Apr-2020.)
(𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
 
Theoremtermoo 17258 A terminal object is an object. (Contributed by AV, 18-Apr-2020.)
(𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
 
Theoremiszeroi 17259 Implication of a class being a zero object. (Contributed by AV, 18-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
 
Theorem2initoinv 17260 Morphisms between two initial objects are inverses. (Contributed by AV, 14-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   (𝜑𝐵 ∈ (InitO‘𝐶))       ((𝜑𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)
 
Theoreminitoeu1 17261* Initial objects are essentially unique (strong form), i.e. there is a unique isomorphism between two initial objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 14-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   (𝜑𝐵 ∈ (InitO‘𝐶))       (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
 
Theoreminitoeu1w 17262 Initial objects are essentially unique (weak form), i.e. if A and B are initial objects, then A and B are isomorphic. Proposition 7.3 (1) of [Adamek] p. 102. (Contributed by AV, 6-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   (𝜑𝐵 ∈ (InitO‘𝐶))       (𝜑𝐴( ≃𝑐𝐶)𝐵)
 
Theoreminitoeu2lem0 17263 Lemma 0 for initoeu2 17266. (Contributed by AV, 9-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   𝑋 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &    = (comp‘𝐶)       (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) ∧ ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾)(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
 
Theoreminitoeu2lem1 17264* Lemma 1 for initoeu2 17266. (Contributed by AV, 9-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   𝑋 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &    = (comp‘𝐶)       ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))
 
Theoreminitoeu2lem2 17265* Lemma 2 for initoeu2 17266. (Contributed by AV, 10-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   𝑋 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &    = (comp‘𝐶)       ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) → ∃!𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
 
Theoreminitoeu2 17266 Initial objects are essentially unique, if A is an initial object, then so is every object that is isomorphic to A. Proposition 7.3 (2) in [Adamek] p. 102. (Contributed by AV, 10-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   (𝜑𝐴( ≃𝑐𝐶)𝐵)       (𝜑𝐵 ∈ (InitO‘𝐶))
 
Theorem2termoinv 17267 Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (TermO‘𝐶))    &   (𝜑𝐵 ∈ (TermO‘𝐶))       ((𝜑𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)
 
Theoremtermoeu1 17268* Terminal objects are essentially unique (strong form), i.e. there is a unique isomorphism between two terminal objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 18-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (TermO‘𝐶))    &   (𝜑𝐵 ∈ (TermO‘𝐶))       (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
 
Theoremtermoeu1w 17269 Terminal objects are essentially unique (weak form), i.e. if A and B are terminal objects, then A and B are isomorphic. Proposition 7.6 of [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (TermO‘𝐶))    &   (𝜑𝐵 ∈ (TermO‘𝐶))       (𝜑𝐴( ≃𝑐𝐶)𝐵)
 
8.2  Arrows (disjointified hom-sets)
 
Syntaxcdoma 17270 Extend class notation to include the domain extractor for an arrow.
class doma
 
Syntaxccoda 17271 Extend class notation to include the codomain extractor for an arrow.
class coda
 
Syntaxcarw 17272 Extend class notation to include the collection of all arrows of a category.
class Arrow
 
Syntaxchoma 17273 Extend class notation to include the set of all arrows with a specific domain and codomain.
class Homa
 
Definitiondf-doma 17274 Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
doma = (1st ∘ 1st )
 
Definitiondf-coda 17275 Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
coda = (2nd ∘ 1st )
 
Definitiondf-homa 17276* Definition of the hom-set extractor for arrows, which tags the morphisms of the underlying hom-set with domain and codomain, which can then be extracted using df-doma 17274 and df-coda 17275. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
 
Definitiondf-arw 17277 Definition of the set of arrows of a category. We will use the term "arrow" to denote a morphism tagged with its domain and codomain, as opposed to Hom, which allows hom-sets for distinct objects to overlap. (Contributed by Mario Carneiro, 11-Jan-2017.)
Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
 
Theoremhomarcl 17278 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
 
Theoremhomafval 17279* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
 
Theoremhomaf 17280 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
 
Theoremhomaval 17281 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
 
Theoremelhoma 17282 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
 
Theoremelhomai 17283 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐽𝑌))       (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)
 
Theoremelhomai2 17284 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐽𝑌))       (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))
 
Theoremhomarcl2 17285 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋𝐵𝑌𝐵))
 
Theoremhomarel 17286 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       Rel (𝑋𝐻𝑌)
 
Theoremhoma1 17287 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝑍(𝑋𝐻𝑌)𝐹𝑍 = ⟨𝑋, 𝑌⟩)
 
Theoremhomahom2 17288 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐽 = (Hom ‘𝐶)       (𝑍(𝑋𝐻𝑌)𝐹𝐹 ∈ (𝑋𝐽𝑌))
 
Theoremhomahom 17289 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐽 = (Hom ‘𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋𝐽𝑌))
 
Theoremhomadm 17290 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
 
Theoremhomacd 17291 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
 
Theoremhomadmcd 17292 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
 
Theoremarwval 17293 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐻 = (Homa𝐶)       𝐴 = ran 𝐻
 
Theoremarwrcl 17294 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)       (𝐹𝐴𝐶 ∈ Cat)
 
Theoremarwhoma 17295 An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐻 = (Homa𝐶)       (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
 
Theoremhomarw 17296 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐻 = (Homa𝐶)       (𝑋𝐻𝑌) ⊆ 𝐴
 
Theoremarwdm 17297 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹𝐴 → (doma𝐹) ∈ 𝐵)
 
Theoremarwcd 17298 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹𝐴 → (coda𝐹) ∈ 𝐵)
 
Theoremdmaf 17299 The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (doma𝐴):𝐴𝐵
 
Theoremcdaf 17300 The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (coda𝐴):𝐴𝐵
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