P. 505 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50401-50500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	upfval2 50401*	Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 𝑂 = (comp‘𝐸)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))})
Theorem	upfval3 50402*	Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 𝑂 = (comp‘𝐸)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → (⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚))})
Theorem	isuplem 50403*	Lemma for isup 50404 and other theorems. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 𝑂 = (comp‘𝐸)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))))
Theorem	isup 50404*	The predicate "is a universal pair". (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 𝑂 = (comp‘𝐸)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))    ⇒   ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀)))
Theorem	uppropd 50405	If two categories have the same set of objects, morphisms, and compositions, then they have the same universal pairs. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 UP 𝐶) = (𝐵 UP 𝐷))
Theorem	reldmup2 50406	The domain of (𝐷 UP 𝐸) is a relation. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ Rel dom (𝐷 UP 𝐸)
Theorem	relup 50407	The set of universal pairs is a relation. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ Rel (𝐹(𝐷 UP 𝐸)𝑊)
Theorem	uprcl 50408	Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ 𝐶 = (Base‘𝐸)    ⇒   ⊢ (𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ 𝐶))
Theorem	up1st2nd 50409	Rewrite the universal property predicate with separated parts. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)    ⇒   ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)
Theorem	up1st2ndr 50410	Combine separated parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)    ⇒   ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
Theorem	up1st2ndb 50411	Combine/separate parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀))
Theorem	up1st2nd2 50412	Rewrite the universal property predicate with separated parts. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ (𝜑 → 𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊))    ⇒   ⊢ (𝜑 → (1st ‘𝑋)(𝐹(𝐷 UP 𝐸)𝑊)(2nd ‘𝑋))
Theorem	uprcl2 50413	Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)    ⇒   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
Theorem	uprcl3 50414	Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)    &   ⊢ 𝐶 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → 𝑊 ∈ 𝐶)
Theorem	uprcl4 50415	Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)    &   ⊢ 𝐵 = (Base‘𝐷)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐵)
Theorem	uprcl5 50416	Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)    &   ⊢ 𝐽 = (Hom ‘𝐸)    ⇒   ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))
Theorem	uobrcl 50417	Reverse closure for universal object. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
Theorem	isup2 50418*	The universal property of a universal pair. (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 𝑂 = (comp‘𝐸)    &   ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)    ⇒   ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))
Theorem	upeu3 50419*	The universal pair ⟨𝑋, 𝑀⟩ from object 𝑊 to functor ⟨𝐹, 𝐺⟩ is essentially unique (strong form) if it exists. (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ (𝜑 → 𝐼 = (Iso‘𝐷))    &   ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))    &   ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)    &   ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)    ⇒   ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋𝐼𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀))
Theorem	upeu4 50420	Generate a new universal morphism through an isomorphism from an existing universal object, and pair with the codomain of the isomorphism to form a universal pair. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ (𝜑 → 𝐼 = (Iso‘𝐷))    &   ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))    &   ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))    &   ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾) ⚬ 𝑀))    ⇒   ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)
Theorem	uptposlem 50421	Lemma for uptpos 50422. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ (𝜑 → tpos 𝐺 = 𝐻)    ⇒   ⊢ (𝜑 → tpos 𝐻 = 𝐺)
Theorem	uptpos 50422	Rewrite the predicate of universal property in the form of opposite functor. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ (𝜑 → tpos 𝐺 = 𝐻)    ⇒   ⊢ (𝜑 → 𝑋(⟨𝐹, tpos 𝐻⟩(𝑂 UP 𝑃)𝑊)𝑀)
Theorem	oppcuprcl4 50423	Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ 𝑂 = (oppCat‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐵)
Theorem	oppcuprcl3 50424	Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ 𝑃 = (oppCat‘𝐸)    &   ⊢ 𝐶 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → 𝑊 ∈ 𝐶)
Theorem	oppcuprcl5 50425	Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ 𝑃 = (oppCat‘𝐸)    &   ⊢ 𝐽 = (Hom ‘𝐸)    ⇒   ⊢ (𝜑 → 𝑀 ∈ ((𝐹‘𝑋)𝐽𝑊))
Theorem	oppcuprcl2 50426	Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ 𝑃 = (oppCat‘𝐸)    &   ⊢ 𝑂 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → tpos 𝐺 = 𝐻)    ⇒   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐻)
Theorem	uprcl2a 50427	Reverse closure for the class of universal property. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀)    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑂 Func 𝑃))
Theorem	oppfuprcl 50428	Reverse closure for the class of universal property for opposite functors. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    &   ⊢ 𝑂 = (oppCat‘𝐷)    &   ⊢ 𝑃 = (oppCat‘𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
Theorem	oppfuprcl2 50429	Reverse closure for the class of universal property for opposite functors. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    &   ⊢ 𝑂 = (oppCat‘𝐷)    &   ⊢ 𝑃 = (oppCat‘𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)    ⇒   ⊢ (𝜑 → 𝐴(𝐷 Func 𝐸)𝐵)
Theorem	oppcup3lem 50430*	Lemma for oppcup3 50433. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ ((𝐹‘𝑦)𝐽𝑍)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘)))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑍))    ⇒   ⊢ (𝜑 → ∃!𝑙 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙)))
Theorem	oppcup 50431*	The universal pair ⟨𝑋, 𝑀⟩ from a functor to an object is universal from an object to a functor in the opposite category. (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ ∙ = (comp‘𝐸)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ ((𝐹‘𝑋)𝐽𝑊))    &   ⊢ 𝑂 = (oppCat‘𝐷)    &   ⊢ 𝑃 = (oppCat‘𝐸)    ⇒   ⊢ (𝜑 → (𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘))))
Theorem	oppcup2 50432*	The universal property for the universal pair ⟨𝑋, 𝑀⟩ from a functor to an object, expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ ∙ = (comp‘𝐸)    &   ⊢ 𝑂 = (oppCat‘𝐷)    &   ⊢ 𝑃 = (oppCat‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀)    ⇒   ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))
Theorem	oppcup3 50433*	The universal property for the universal pair ⟨𝑋, 𝑀⟩ from a functor to an object, expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ ∙ = (comp‘𝐸)    &   ⊢ 𝑂 = (oppCat‘𝐷)    &   ⊢ 𝑃 = (oppCat‘𝐸)    &   ⊢ (𝜑 → 𝑋(⟨𝐹, 𝑇⟩(𝑂 UP 𝑃)𝑊)𝑀)    &   ⊢ (𝜑 → tpos 𝑇 = 𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑊))    ⇒   ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑌𝐺𝑋)‘𝑘)))
Theorem	uptrlem1 50434*	Lemma for uptr 50437. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ ⚬ = (comp‘𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → (𝑀‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝐹‘𝑍)))    &   ⊢ (𝜑 → ((𝑋𝑁(𝐹‘𝑍))‘𝐴) = 𝐵)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑀((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑁)    &   ⊢ (𝜑 → (⟨𝑀, 𝑁⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)    ⇒   ⊢ (𝜑 → (∀ℎ ∈ (𝑌𝐽(𝐾‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)ℎ = (((𝑍𝐿𝑊)‘𝑘)(⟨𝑌, (𝐾‘𝑍)⟩ ⚬ (𝐾‘𝑊))𝐵) ↔ ∀𝑔 ∈ (𝑋𝐼(𝐹‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)𝑔 = (((𝑍𝐺𝑊)‘𝑘)(⟨𝑋, (𝐹‘𝑍)⟩ ∙ (𝐹‘𝑊))𝐴)))
Theorem	uptrlem2 50435*	Lemma for uptr 50437. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ ⚬ = (comp‘𝐸)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼((1st ‘𝐹)‘𝑍)))    &   ⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    ⇒   ⊢ (𝜑 → (∀ℎ ∈ (𝑌𝐽((1st ‘𝐺)‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)ℎ = (((𝑍(2nd ‘𝐺)𝑊)‘𝑘)(⟨𝑌, ((1st ‘𝐺)‘𝑍)⟩ ⚬ ((1st ‘𝐺)‘𝑊))𝑁) ↔ ∀𝑔 ∈ (𝑋𝐼((1st ‘𝐹)‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)𝑔 = (((𝑍(2nd ‘𝐹)𝑊)‘𝑘)(⟨𝑋, ((1st ‘𝐹)‘𝑍)⟩ ∙ ((1st ‘𝐹)‘𝑊))𝑀)))
Theorem	uptrlem3 50436	Lemma for uptr 50437. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ (𝜑 → (𝑅‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)    &   ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))
Theorem	uptr 50437	Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ (𝜑 → (𝑅‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)    &   ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))    ⇒   ⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))
Theorem	uptri 50438	Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ (𝜑 → (𝑅‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)    &   ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)    &   ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)    &   ⊢ (𝜑 → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)    ⇒   ⊢ (𝜑 → 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁)
Theorem	uptra 50439	Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽((1st ‘𝐹)‘𝑍)))    ⇒   ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
Theorem	uptrar 50440	Universal property and fully faithful functor. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) = 𝑀)    &   ⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)    ⇒   ⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)
Theorem	uptrai 50441	Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   ⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)    &   ⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)    ⇒   ⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)
Theorem	uobffth 50442	A fully faithful functor generates equal sets of universal objects. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    ⇒   ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
Theorem	uobeqw 50443	If a full functor (in fact, a full embedding) is a section of a fully faithful functor (surjective on objects), then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))    &   ⊢ (𝜑 → (𝐿 ∘func 𝐾) = 𝐼)    &   ⊢ (𝜑 → 𝐿 ∈ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)))    ⇒   ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
Theorem	uobeq 50444	If a full functor (in fact, a full embedding) is a section of a functor (surjective on objects), then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))    &   ⊢ (𝜑 → (𝐿 ∘func 𝐾) = 𝐼)    &   ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))    ⇒   ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
Theorem	uptr2 50445	Universal property and fully faithful functor surjective on objects. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑌 = (𝑅‘𝑋))    &   ⊢ (𝜑 → 𝑅:𝐴–onto→𝐵)    &   ⊢ (𝜑 → 𝑅((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝑆)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝑅, 𝑆⟩) = ⟨𝐹, 𝐺⟩)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    ⇒   ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀))
Theorem	uptr2a 50446	Universal property and fully faithful functor surjective on objects. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑌 = ((1st ‘𝐾)‘𝑋))    &   ⊢ (𝜑 → (𝐺 ∘func 𝐾) = 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐾 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))    &   ⊢ (𝜑 → (1st ‘𝐾):𝐴–onto→𝐵)    ⇒   ⊢ (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(𝐺(𝐷 UP 𝐸)𝑍)𝑀))
21.53.15.11  Natural transformations and the functor category
Theorem	isnatd 50447*	Property of being a natural transformation; deduction form. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ · = (comp‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)    &   ⊢ (𝜑 → 𝐴 Fn 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))    &   ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ℎ ∈ (𝑥𝐻𝑦)) → ((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
Theorem	natrcl2 50448	Reverse closure for a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))    ⇒   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
Theorem	natrcl3 50449	Reverse closure for a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))    ⇒   ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
Theorem	isnat3 50450*	A natural transformation is an exact component function with typed values satisfying naturality. (Contributed by Zhi Wang, 14-Sep-2026.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ · = (comp‘𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1st ‘𝐹)‘𝑥)𝐽((1st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩ · ((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝐺)𝑦)‘ℎ)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))
Theorem	natfn2 50451	A natural transformation is a function on the objects of the source category. (Contributed by Zhi Wang, 14-Sep-2026.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    ⇒   ⊢ (𝜑 → 𝐴 Fn 𝐵)
Theorem	nateq 50452*	Two natural transformations with the same source and target are equal exactly when all their components agree. (Contributed by Zhi Wang, 14-Sep-2026.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹𝑁𝐺))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐿 ↔ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) = (𝐿‘𝑥)))
Theorem	natideq 50453*	A natural transformation is an identity exactly when all its components are identities. This also covers an empty object set. (Contributed by Zhi Wang, 14-Sep-2026.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ 𝐽 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐹))    ⇒   ⊢ (𝜑 → (𝐴 = (𝐼‘𝐹) ↔ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) = (𝐽‘((1st ‘𝐹)‘𝑥))))
Theorem	natcoeq 50454*	A natural transformation equals a composite exactly when all its components equal the corresponding composites, including when the object set is empty. (Contributed by Zhi Wang, 14-Sep-2026.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ⚬ = (comp‘𝐷)    &   ⊢ · = (comp‘𝑄)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐺𝑁𝐻))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹𝑁𝐻))    ⇒   ⊢ (𝜑 → (𝐿 = (𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴) ↔ ∀𝑥 ∈ 𝐵 (𝐿‘𝑥) = ((𝑅‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ ⚬ ((1st ‘𝐻)‘𝑥))(𝐴‘𝑥))))
Theorem	fucidcobi 50455*	Componentwise functor laws for a correctly typed hom-family into a functor category. The source category 𝑇 and object assignment 𝑂 are arbitrary: the identity and composition laws are equivalent to their component equations. No functor law is assumed in the hypotheses. Functor composition is an application in fucoisfunc2 50598. (Contributed by Zhi Wang, 14-Sep-2026.)
⊢ 𝑊 = (Base‘𝑇)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ 𝐼 = (Id‘𝑇)    &   ⊢ 𝐽 = (Id‘𝑄)    &   ⊢ 𝐾 = (Id‘𝐸)    &   ⊢ · = (comp‘𝑇)    &   ⊢ ⚬ = (comp‘𝑄)    &   ⊢ ∙ = (comp‘𝐸)    &   ⊢ (𝜑 → 𝑇 ∈ Cat)    &   ⊢ (𝜑 → 𝑃 ∈ X𝑣 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑣))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑣))) ↑m (𝐻‘𝑣)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))) ↔ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡)))))
Theorem	catbas 50456	The base of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ 𝐵 = (Base‘𝐶)
Theorem	cathomfval 50457	The hom-sets of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}    &   ⊢ 𝐻 ∈ V    ⇒   ⊢ 𝐻 = (Hom ‘𝐶)
Theorem	catcofval 50458	Composition of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}    &   ⊢ · ∈ V    ⇒   ⊢ · = (comp‘𝐶)
Theorem	natoppf 50459	A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝑀 = (𝑂 Nat 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩))
Theorem	natoppf2 50460	A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝑀 = (𝑂 Nat 𝑃)    &   ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))    &   ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐿𝑀𝐾))
Theorem	natoppfb 50461	A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝑀 = (𝑂 Nat 𝑃)    &   ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))    &   ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))
21.53.15.12  Initial, terminal and zero objects of a category
Theorem	initoo2 50462	An initial object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵)
Theorem	termoo2 50463	A terminal object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵)
Theorem	zeroo2 50464	A zero object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝑂 ∈ (ZeroO‘𝐶) → 𝑂 ∈ 𝐵)
Theorem	oppcinito 50465	Initial objects are terminal in the opposite category. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ (TermO‘(oppCat‘𝐶)))
Theorem	oppctermo 50466	Terminal objects are initial in the opposite category. Comments before Definition 7.4 in [Adamek] p. 102. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ (InitO‘(oppCat‘𝐶)))
Theorem	oppczeroo 50467	Zero objects are zero in the opposite category. Remark 7.8 of [Adamek] p. 103. (Contributed by Zhi Wang, 27-Oct-2025.)
⊢ (𝐼 ∈ (ZeroO‘𝐶) ↔ 𝐼 ∈ (ZeroO‘(oppCat‘𝐶)))
Theorem	termoeu2 50468	Terminal objects are essentially unique; if 𝐴 is a terminal object, then so is every object that is isomorphic to 𝐴. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))    &   ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))
Theorem	initopropdlemlem 50469	Lemma for initopropdlem 50470, termopropdlem 50471, and zeroopropdlem 50472. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ 𝐹 Fn 𝑋    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝑌)    &   ⊢ 𝑋 ⊆ 𝑌    &   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))
Theorem	initopropdlem 50470	Lemma for initopropd 50473. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → ¬ 𝐶 ∈ V)    ⇒   ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))
Theorem	termopropdlem 50471	Lemma for termopropd 50474. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → ¬ 𝐶 ∈ V)    ⇒   ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))
Theorem	zeroopropdlem 50472	Lemma for zeroopropd 50475. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → ¬ 𝐶 ∈ V)    ⇒   ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))
Theorem	initopropd 50473	Two structures with the same base, hom-sets and composition operation have the same initial objects. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷))
Theorem	termopropd 50474	Two structures with the same base, hom-sets and composition operation have the same terminal objects. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))
Theorem	zeroopropd 50475	Two structures with the same base, hom-sets and composition operation have the same zero objects. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))
21.53.15.13  Product of categories
Theorem	reldmxpc 50476	The binary product of categories is a proper operator, so it can be used with ovprc1 7463, elbasov 17403, strov2rcl 17404, and so on. See reldmxpcALT 50477 for an alternate proof with less "essential steps" but more "bytes". (Proposed by SN, 15-Oct-2025.) (Contributed by Zhi Wang, 15-Oct-2025.)
⊢ Rel dom ×c
Theorem	reldmxpcALT 50477	Alternate proof of reldmxpc 50476. (Contributed by Zhi Wang, 15-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Rel dom ×c
Theorem	elxpcbasex1 50478	A non-empty base set of the product category indicates the existence of the first factor of the product category. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof shortened by SN, 15-Oct-2025.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)
Theorem	elxpcbasex1ALT 50479	Alternate proof of elxpcbasex1 50478. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)
Theorem	elxpcbasex2 50480	A non-empty base set of the product category indicates the existence of the second factor of the product category. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof shortened by SN, 15-Oct-2025.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)
Theorem	elxpcbasex2ALT 50481	Alternate proof of elxpcbasex2 50480. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)
Theorem	xpcfucbas 50482	The base set of the product of two categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    ⇒   ⊢ ((𝐵 Func 𝐶) × (𝐷 Func 𝐸)) = (Base‘𝑇)
Theorem	xpcfuchomfval 50483*	Set of morphisms of the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝐴 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ 𝐾 = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐴 ↦ (((1st ‘𝑢)(𝐵 Nat 𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(𝐷 Nat 𝐸)(2nd ‘𝑣))))
Theorem	xpcfuchom 50484	Set of morphisms of the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝐴 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)(𝐵 Nat 𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(𝐷 Nat 𝐸)(2nd ‘𝑌))))
Theorem	xpcfuchom2 50485	Value of the set of morphisms in the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ (𝜑 → 𝑀 ∈ (𝐵 Func 𝐶))    &   ⊢ (𝜑 → 𝑁 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑃 ∈ (𝐵 Func 𝐶))    &   ⊢ (𝜑 → 𝑄 ∈ (𝐷 Func 𝐸))    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ (𝜑 → (⟨𝑀, 𝑁⟩𝐾⟨𝑃, 𝑄⟩) = ((𝑀(𝐵 Nat 𝐶)𝑃) × (𝑁(𝐷 Nat 𝐸)𝑄)))
Theorem	xpcfucco2 50486	Value of composition in the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺)⟩)
Theorem	xpcfuccocl 50487	The composition of two natural transformations is a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) ∈ ((𝑀(𝐵 Nat 𝐶)𝑅) × (𝑁(𝐷 Nat 𝐸)𝑆)))
Theorem	xpcfucco3 50488*	Value of composition in the binary product of categories of functors; expressed explicitly. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))    &   ⊢ 𝑋 = (Base‘𝐵)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐸)    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝑥 ∈ 𝑋 ↦ ((𝐾‘𝑥)(⟨((1st ‘𝑀)‘𝑥), ((1st ‘𝑃)‘𝑥)⟩ · ((1st ‘𝑅)‘𝑥))(𝐹‘𝑥))), (𝑦 ∈ 𝑌 ↦ ((𝐿‘𝑦)(⟨((1st ‘𝑁)‘𝑦), ((1st ‘𝑄)‘𝑦)⟩ ∙ ((1st ‘𝑆)‘𝑦))(𝐺‘𝑦)))⟩)
21.53.15.14  Swap functors
Syntax	cswapf 50489	Extend class notation with the class of swap functors.
class swapF
Definition	df-swapf 50490*	Define the swap functor from (𝐶 ×c 𝐷) to (𝐷 ×c 𝐶) by swapping all objects (swapf1 50502) and morphisms (swapf2 50504) . Such functor is called a "swap functor" in https://arxiv.org/pdf/2302.07810 50504 or a "twist functor" in https://arxiv.org/pdf/2508.01886 50504, the latter of which finds its counterpart as "twisting map" in https://arxiv.org/pdf/2411.04102 50504 for tensor product of algebras. The "swap functor" or "twisting map" is often denoted as a small tau 𝜏 in literature. However, the term "twist functor" is defined differently in https://arxiv.org/pdf/1208.4046 50504 and thus not adopted here. tpos I depends on more mathbox theorems, and thus are not adopted here. See dfswapf2 50491 for an alternate definition. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
Theorem	dfswapf2 50491*	Alternate definition of swapF (df-swapf 50490). (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
Theorem	swapfval 50492*	Value of the swap functor. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    ⇒   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
Theorem	swapfelvv 50493	A swap functor is an ordered pair. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
Theorem	swapf2fvala 50494*	The morphism part of the swap functor. See also swapf2fval 50495. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    ⇒   ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
Theorem	swapf2fval 50495*	The morphism part of the swap functor. See also swapf2fvala 50494. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
Theorem	swapf1vala 50496*	The object part of the swap functor. See also swapf1val 50497. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
Theorem	swapf1val 50497*	The object part of the swap functor. See also swapf1vala 50496. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → 𝑂 = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
Theorem	swapf2fn 50498	The morphism part of the swap functor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵))
Theorem	swapf1a 50499	The object part of the swap functor swaps the objects. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩)
Theorem	swapf2vala 50500*	The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    ⇒   ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))