P. 501 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50001-50100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	diagffth 50001	The diagonal functor is a fully faithful functor from a category 𝐶 to the category of functors from a terminal category to 𝐶. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐶)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    ⇒   ⊢ (𝜑 → 𝐿 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
Theorem	diagciso 50002	The diagonal functor is an isomorphism from a category 𝐶 to the category of functors from a terminal category to 𝐶. It is provable that the inverse of the diagonal functor is the mapped object by the transposed curry of (𝐷 evalF 𝐶), i.e., ∪ ran (1st ‘(⟨𝐷, 𝑄⟩ curryF ((𝐷 evalF 𝐶) ∘func (𝐷 swapF 𝑄)))). (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐶)    &   ⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑈)    &   ⊢ 𝐼 = (Iso‘𝐸)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (𝐶𝐼𝑄))
Theorem	diagcic 50003	Any category 𝐶 is isomorphic to the category of functors from a terminal category to 𝐶. See also the "Properties" section of https://ncatlab.org/nlab/show/terminal+category. Therefore the number of categories isomorphic to a non-empty category is at least the number of singletons, so large (snnex 7701) that these isomorphic categories form a proper class. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐶)    &   ⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝐶( ≃𝑐 ‘𝐸)𝑄)
Theorem	funcsn 50004	The category of one functor to a thin category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐶 Func 𝐷) = {𝐹})    &   ⊢ (𝜑 → 𝐷 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝑄 ∈ TermCat)
Theorem	fucterm 50005	The category of functors to a terminal category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    ⇒   ⊢ (𝜑 → 𝑄 ∈ TermCat)
Theorem	0fucterm 50006	The category of functors from an initial category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ∅ = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    ⇒   ⊢ (𝜑 → 𝑄 ∈ TermCat)
Theorem	termfucterm 50007	All functors between two terminal categories are isomorphisms. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ TermCat)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ TermCat)    ⇒   ⊢ (𝜑 → (𝑋 Func 𝑌) = (𝑋𝐼𝑌))
Theorem	cofuterm 50008	Post-compose with a functor to a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ TermCat)    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐾)
Theorem	uobeqterm 50009	Universal objects and terminal categories. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐴 = (Base‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝐸 ∈ TermCat)    ⇒   ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
Theorem	isinito4 50010	The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 1 ∈ TermCat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘ 1 ))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))    ⇒   ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ dom (𝐹(𝐶 UP 1 )𝑋)))
Theorem	isinito4a 50011	The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 1 ∈ TermCat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘ 1 ))    &   ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘𝑋)    ⇒   ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ dom (𝐹(𝐶 UP 1 )𝑋)))
21.49.16.4  Preordered sets as thin categories
Syntax	cprstc 50012	Class function defining preordered sets as categories.
class ProsetToCat
Definition	df-prstc 50013	Definition of the function converting a preordered set to a category. Justified by prsthinc 49927. This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism (thincciso 49916), by prstcnid 50016, prstchom 50025, and prstcthin 50024. Other important properties include prstcbas 50017, prstcleval 50018, prstcle 50019, prstcocval 50020, prstcoc 50021, prstchom2 50026, and prstcprs 50023. Use those instead. Note that the defining property prstchom 50025 is equivalent to prstchom2 50026 given prstcthin 50024. See thincn0eu 49894 for justification. "ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Theorem	prstcval 50014	Lemma for prstcnidlem 50015 and prstcthin 50024. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Theorem	prstcnidlem 50015	Lemma for prstcnid 50016 and prstchomval 50022. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (comp‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
Theorem	prstcnid 50016	Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (comp‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶))
Theorem	prstcbas 50017	The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐾))    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
Theorem	prstcleval 50018	Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    ⇒   ⊢ (𝜑 → ≤ = (le‘𝐶))
Theorem	prstcle 50019	Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐶)𝑌))
Theorem	prstcocval 50020	Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ⊥ = (oc‘𝐾))    ⇒   ⊢ (𝜑 → ⊥ = (oc‘𝐶))
Theorem	prstcoc 50021	Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ⊥ = (oc‘𝐾))    ⇒   ⊢ (𝜑 → ( ⊥ ‘𝑋) = ((oc‘𝐶)‘𝑋))
Theorem	prstchomval 50022	Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    ⇒   ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))
Theorem	prstcprs 50023	The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 ∈ Proset )
Theorem	prstcthin 50024	The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	prstchom 50025	Hom-sets of the constructed category are dependent on the preorder. Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
Theorem	prstchom2 50026*	Hom-sets of the constructed category are dependent on the preorder. Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat ( see prstchom2ALT 50027). However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	prstchom2ALT 50027*	Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 50013. See prstchom2 50026 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	oduoppcbas 50028	The dual of a preordered set and the opposite category have the same set of objects. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → 𝐷 = (ProsetToCat‘(ODual‘𝐾)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝑂))
Theorem	oduoppcciso 50029	The dual of a preordered set and the opposite category are category-isomorphic. Example 3.6(1) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → 𝐷 = (ProsetToCat‘(ODual‘𝐾)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝐷( ≃𝑐 ‘(CatCat‘𝑈))𝑂)
Theorem	postcpos 50030	The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
Theorem	postcposALT 50031	Alternate proof of postcpos 50030. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
Theorem	postc 50032*	The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦)))
Theorem	discsntermlem 50033*	A singlegon is an element of the class of singlegons. The converse (basrestermcfolem 50034) also holds. This is trivial if 𝐵 is 𝑏 (abid 2717). (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}})
Theorem	basrestermcfolem 50034*	An element of the class of singlegons is a singlegon. The converse (discsntermlem 50033) also holds. This is trivial if 𝐵 is 𝑏 (abid 2717). (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → ∃𝑥 𝐵 = {𝑥})
Theorem	discbas 50035	A discrete category (a category whose only morphisms are the identity morphisms) can be constructed for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   ⊢ 𝐶 = (ProsetToCat‘𝐾)    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐶))
Theorem	discthin 50036	A discrete category (a category whose only morphisms are the identity morphisms) is thin. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   ⊢ 𝐶 = (ProsetToCat‘𝐾)    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐶 ∈ ThinCat)
Theorem	discsnterm 50037*	A discrete category (a category whose only morphisms are the identity morphisms) with a singlegon base is terminal. Corollary of example 3.3(4)(c) of [Adamek] p. 24 and example 3.26(1) of [Adamek] p. 33. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   ⊢ 𝐶 = (ProsetToCat‘𝐾)    ⇒   ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐶 ∈ TermCat)
Theorem	basrestermcfo 50038*	The base function restricted to the class of terminal categories maps the class of terminal categories onto the class of singletons. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (Base ↾ TermCat):TermCat–onto→{𝑏 ∣ ∃𝑥 𝑏 = {𝑥}}
Theorem	termcnex 50039	The class of all terminal categories is a proper class. Therefore both the class of all thin categories and the class of all categories are proper classes. Note that snnex 7701 is equivalent to sngl V ∉ V. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ TermCat ∉ V
21.49.16.5  Monoids as categories
Syntax	cmndtc 50040	Class function defining monoids as categories.
class MndToCat
Definition	df-mndtc 50041	Definition of the function converting a monoid to a category. Example 3.3(4.e) of [Adamek] p. 24. The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 50043), instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24. The resulting category is defined entirely, up to isomorphism, by mndtcbas 50044, mndtchom 50047, mndtcco 50048. Use those instead. See example 3.26(3) of [Adamek] p. 33 for more on isomorphism. "MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
Theorem	mndtcval 50042	Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
Theorem	mndtcbasval 50043	The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = {𝑀})
Theorem	mndtcbas 50044*	The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵)
Theorem	mndtcob 50045	Lemma for mndtchom 50047 and mndtcco 50048. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑀)
Theorem	mndtcbas2 50046	Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	mndtchom 50047	The only hom-set of the category built from a monoid is the base set of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (Proof shortened by Zhi Wang, 22-Oct-2025.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (Base‘𝑀))
Theorem	mndtcco 50048	The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → · = (comp‘𝐶))    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (+g‘𝑀))
Theorem	mndtcco2 50049	The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝜑 → ⚬ = (⟨𝑋, 𝑌⟩ · 𝑍))    ⇒   ⊢ (𝜑 → (𝐺 ⚬ 𝐹) = (𝐺(+g‘𝑀)𝐹))
Theorem	mndtccatid 50050*	Lemma for mndtccat 50051 and mndtcid 50052. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))
Theorem	mndtccat 50051	The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	mndtcid 50052	The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 1 = (Id‘𝐶))    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = (0g‘𝑀))
Theorem	oppgoppchom 50053	The converted opposite monoid has the same hom-set as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 21-Sep-2025.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐷))    &   ⊢ (𝜑 → 𝐽 = (Hom ‘𝑂))    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑌𝐽𝑌))
Theorem	oppgoppcco 50054	The converted opposite monoid has the same composition as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))    &   ⊢ (𝜑 → · = (comp‘𝐷))    &   ⊢ (𝜑 → ∙ = (comp‘𝑂))    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (⟨𝑌, 𝑌⟩ ∙ 𝑌))
Theorem	oppgoppcid 50055	The converted opposite monoid has the same identity morphism as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))    ⇒   ⊢ (𝜑 → ((Id‘𝐷)‘𝑋) = ((Id‘𝑂)‘𝑌))
Theorem	grptcmon 50056	All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝑀 = (Mono‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
Theorem	grptcepi 50057	All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝐸 = (Epi‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
21.49.16.6  Categories with at most one object and at most two morphisms
Theorem	2arwcatlem1 50058	Lemma for 2arwcat 50063. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝑋𝐻𝑋) = { 0 , 1 }    ⇒   ⊢ ((((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 ))) ↔ ((𝑥 ∈ {𝑋} ∧ 𝑦 ∈ {𝑋}) ∧ (𝑧 ∈ {𝑋} ∧ 𝑤 ∈ {𝑋}) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
Theorem	2arwcatlem2 50059	Lemma for 2arwcat 50063. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝑋)    &   ⊢ (𝜑 → 𝐵 = 𝑌)    &   ⊢ (𝜑 → 𝐶 = 𝑍)    &   ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )    ⇒   ⊢ (𝜑 → ( 1 (⟨𝐴, 𝐵⟩ · 𝐶)𝐹) = 𝐹)
Theorem	2arwcatlem3 50060	Lemma for 2arwcat 50063. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝑋)    &   ⊢ (𝜑 → 𝐵 = 𝑌)    &   ⊢ (𝜑 → 𝐶 = 𝑍)    &   ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )    ⇒   ⊢ (𝜑 → (𝐹(⟨𝐴, 𝐵⟩ · 𝐶) 1 ) = 𝐹)
Theorem	2arwcatlem4 50061	Lemma for 2arwcat 50063. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝑋)    &   ⊢ (𝜑 → 𝐵 = 𝑌)    &   ⊢ (𝜑 → 𝐶 = 𝑍)    &   ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ { 0 , 1 })    &   ⊢ (𝜑 → (𝐺 = 0 ∨ 𝐺 = 1 ))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝐴, 𝐵⟩ · 𝐶)𝐹) ∈ { 0 , 1 })
Theorem	2arwcatlem5 50062	Lemma for 2arwcat 50063. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → ( 1 · 0 ) = 0 )    &   ⊢ (𝜑 → ( 0 · 1 ) = 0 )    &   ⊢ (𝜑 → ( 0 · 0 ) ∈ { 0 , 1 })    ⇒   ⊢ (𝜑 → (( 0 · 0 ) · 0 ) = ( 0 · ( 0 · 0 )))
Theorem	2arwcat 50063*	The condition for a structure with at most one object and at most two morphisms being a category. "2arwcat.2" to "2arwcat.5" are also necessary conditions if 𝑋, 0, and 1 are all sets, due to catlid 17638, catrid 17639, and catcocl 17640. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → {𝑋} = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝑋𝐻𝑋) = { 0 , 1 }    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })    ⇒   ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 1 )))
Theorem	incat 50064*	Constructing a category with at most one object and at most two morphisms. If 𝑋 is a set then 𝐶 is the category 𝐴 in Exercise 3G of [Adamek] p. 45. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐶 = {⟨(Base‘ndx), {𝑋}⟩, ⟨(Hom ‘ndx), {⟨𝑋, 𝑋, 𝐻⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑋, 𝑋⟩, 𝑋, · ⟩}⟩}    &   ⊢ 𝐻 = {𝐹, 𝐺}    &   ⊢ · = (𝑓 ∈ 𝐻, 𝑔 ∈ 𝐻 ↦ (𝑓 ∩ 𝑔))    ⇒   ⊢ ((𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 𝐺)))
Theorem	setc1onsubc 50065*	Construct a category with one object and two morphisms and prove that category (SetCat‘1o) satisfies all conditions for a subcategory but the compatibility of identity morphisms, showing the necessity of the latter condition in defining a subcategory. Exercise 4A of [Adamek] p. 58. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ 𝐶 = {⟨(Base‘ndx), {∅}⟩, ⟨(Hom ‘ndx), {⟨∅, ∅, 2o⟩}⟩, ⟨(comp‘ndx), {⟨⟨∅, ∅⟩, ∅, · ⟩}⟩}    &   ⊢ · = (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔))    &   ⊢ 𝐸 = (SetCat‘1o)    &   ⊢ 𝐽 = (Homf ‘𝐸)    &   ⊢ 𝑆 = 1o    &   ⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝐷 = (𝐶 ↾cat 𝐽)    ⇒   ⊢ (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
Theorem	cnelsubclem 50066*	Lemma for cnelsubc 50067. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ 𝐽 ∈ V    &   ⊢ 𝑆 ∈ V    &   ⊢ (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat (Homf ‘𝐶) ∧ ¬ ∀𝑥 ∈ 𝑆 ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶 ↾cat 𝐽) ∈ Cat))    ⇒   ⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ ¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ (𝑐 ↾cat 𝑗) ∈ Cat))
Theorem	cnelsubc 50067*	Remark 4.2(2) of [Adamek] p. 48. There exists a category satisfying all conditions for a subcategory but the compatibility of identity morphisms. Therefore such condition in df-subc 17768 is necessary. A stronger statement than nelsubc3 49534. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ ¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ (𝑐 ↾cat 𝑗) ∈ Cat))
21.49.17  Kan extensions and related concepts
21.49.17.1  Kan extensions
Syntax	clan 50068	Class function defining the (local) left Kan extension.
class Lan
Syntax	cran 50069	Class function defining the (local) right Kan extension.
class Ran
Definition	df-lan 50070*	Definition of the (local) left Kan extension. Given a functor 𝐹:𝐶⟶𝐷 and a functor 𝑋:𝐶⟶𝐸, the set (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) consists of left Kan extensions of 𝑋 along 𝐹, which are universal pairs from 𝑋 to the pre-composition functor given by 𝐹 (lanval2 50090). See also § 3 of Chapter X in p. 240 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf 50090 (retrieved 3 Nov 2025). A left Kan extension is in the form of ⟨𝐿, 𝐴⟩ where the first component is a functor 𝐿:𝐷⟶𝐸 (lanrcl4 50097) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 50098) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of an "inverse" of pre-composition; the source category of 𝑋:𝐶⟶𝐸 is "extended" along 𝐹:𝐶⟶𝐷. The left Kan extension is a generalization of many categorical concepts such as colimit. In § 7 of Chapter X of Categories for the Working Mathematician, it is concluded that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory". This definition was chosen over the other version in the commented out section due to its better reverse closure property. See df-ran 50071 for the dual concept. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)))
Definition	df-ran 50071*	Definition of the (local) right Kan extension. Given a functor 𝐹:𝐶⟶𝐷 and a functor 𝑋:𝐶⟶𝐸, the set (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) consists of right Kan extensions of 𝑋 along 𝐹, which are universal pairs from the pre-composition functor given by 𝐹 to 𝑋 (ranval2 50093). The definition in § 3 of Chapter X in p. 236 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf 50093 (retrieved 3 Nov 2025). A right Kan extension is in the form of ⟨𝐿, 𝐴⟩ where the first component is a functor 𝐿:𝐷⟶𝐸 (ranrcl4 50102) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 50103) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of an "inverse" of pre-composition; the source category of 𝑋:𝐶⟶𝐸 is "extended" along 𝐹:𝐶⟶𝐷. The right Kan extension is a generalization of many categorical concepts such as limit. In § 7 of Chapter X of Categories for the Working Mathematician, it is concluded that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory". This definition was chosen over the other version in the commented out section due to its better reverse closure property. See df-lan 50070 for the dual concept. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
Theorem	lanfn 50072	Lan is a function on ((V × V) × V). (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Lan Fn ((V × V) × V)
Theorem	ranfn 50073	Ran is a function on ((V × V) × V). (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Ran Fn ((V × V) × V)
Theorem	reldmlan 50074	The domain of Lan is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel dom Lan
Theorem	reldmran 50075	The domain of Ran is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel dom Ran
Theorem	lanfval 50076*	Value of the function generating the set of left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
Theorem	ranfval 50077*	Value of the function generating the set of right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ 𝑂 = (oppCat‘𝑅)    &   ⊢ 𝑃 = (oppCat‘𝑆)    ⇒   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Ran 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂 UP 𝑃)𝑥)))
Theorem	lanpropd 50078	If the categories have the same set of objects, morphisms, and compositions, then they have the same left Kan extensions. (Contributed by Zhi Wang, 21-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → (Homf ‘𝐸) = (Homf ‘𝐹))    &   ⊢ (𝜑 → (compf‘𝐸) = (compf‘𝐹))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Lan 𝐸) = (⟨𝐵, 𝐷⟩ Lan 𝐹))
Theorem	ranpropd 50079	If the categories have the same set of objects, morphisms, and compositions, then they have the same right Kan extensions. (Contributed by Zhi Wang, 21-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → (Homf ‘𝐸) = (Homf ‘𝐹))    &   ⊢ (𝜑 → (compf‘𝐸) = (compf‘𝐹))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Ran 𝐸) = (⟨𝐵, 𝐷⟩ Ran 𝐹))
Theorem	reldmlan2 50080	The domain of (𝑃 Lan 𝐸) is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel dom (𝑃 Lan 𝐸)
Theorem	reldmran2 50081	The domain of (𝑃 Ran 𝐸) is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel dom (𝑃 Ran 𝐸)
Theorem	lanval 50082	Value of the set of left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝐾)    ⇒   ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
Theorem	ranval 50083	Value of the set of right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   ⊢ 𝑂 = (oppCat‘𝑅)    &   ⊢ 𝑃 = (oppCat‘𝑆)    ⇒   ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
Theorem	lanrcl 50084	Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
Theorem	ranrcl 50085	Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
Theorem	rellan 50086	The set of left Kan extensions is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)
Theorem	relran 50087	The set of right Kan extensions is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel (𝐹(𝑃 Ran 𝐸)𝑋)
Theorem	islan 50088	A left Kan extension is a universal pair. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
Theorem	islan2 50089	A left Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   ⊢ (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 → 𝐿(𝐾(𝑅 UP 𝑆)𝑋)𝐴)
Theorem	lanval2 50090	The set of left Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by isup2 49657 and upciclem1 49629. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
Theorem	isran 50091	A right Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋))    ⇒   ⊢ (𝜑 → 𝐿 ∈ (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
Theorem	isran2 50092	A right Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   ⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐿(⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋)𝐴)
Theorem	ranval2 50093	The set of right Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by oppcup2 49671 and oppcup3lem 49669. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
Theorem	ranval3 50094	The set of right Kan extensions is the set of universal pairs. (Contributed by Zhi Wang, 26-Nov-2025.)
⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))
Theorem	lanrcl2 50095	Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
Theorem	lanrcl3 50096	Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
Theorem	lanrcl4 50097	The first component of a left Kan extension is a functor. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
Theorem	lanrcl5 50098	The second component of a left Kan extension is a natural transformation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    &   ⊢ 𝑁 = (𝐶 Nat 𝐸)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))
Theorem	ranrcl2 50099	Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
Theorem	ranrcl3 50100	Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))