Comma categories are locally small #
We introduce instances showing that the various comma categories are locally small when the relevant categories that are involved are locally small.
instance
CategoryTheory.Comma.locallySmall
{A : Type u₁}
{B : Type u₂}
{T : Type u₃}
[CategoryTheory.Category.{v₁, u₁} A]
[CategoryTheory.Category.{v₂, u₂} B]
[CategoryTheory.Category.{v₃, u₃} T]
(L : CategoryTheory.Functor A T)
(R : CategoryTheory.Functor B T)
[CategoryTheory.LocallySmall.{w, v₁, u₁} A]
[CategoryTheory.LocallySmall.{w, v₂, u₂} B]
:
instance
CategoryTheory.StructuredArrow.locallySmall
{B : Type u₂}
{T : Type u₃}
[CategoryTheory.Category.{v₂, u₂} B]
[CategoryTheory.Category.{v₃, u₃} T]
(S : T)
(T✝ : CategoryTheory.Functor B T)
[CategoryTheory.LocallySmall.{w, v₂, u₂} B]
:
instance
CategoryTheory.CostructuredArrow.locallySmall
{A : Type u₁}
{T : Type u₃}
[CategoryTheory.Category.{v₁, u₁} A]
[CategoryTheory.Category.{v₃, u₃} T]
(S : CategoryTheory.Functor A T)
(X : T)
[CategoryTheory.LocallySmall.{w, v₁, u₁} A]
: