English
The functor map on Coyoneda side commutes with the over-comp Yoneda construction via a naturality relation.
Русский
Граф вывода на стороне Coyoneda commuting с конструкцией overCompYoneda через естественную отношенность.
LaTeX
$$$$ (overEquivPresheafCostructuredArrow A).functor.map (((CostructuredArrow.toOverCompCoyoneda A).inv.app (op X)).app T f) = (CostructuredArrow.toOverCompOverEquivPresheafCostructuredArrow A).hom.app X \\gg f $$$$
Lean4
@[simp]
theorem overEquivPresheafCostructuredArrow_functor_map_toOverCompCoyoneda {A : Cᵒᵖ ⥤ Type v} {T : Over A}
{X : CostructuredArrow yoneda A} (f : yoneda.obj X ⟶ (overEquivPresheafCostructuredArrow A).functor.obj T) :
(overEquivPresheafCostructuredArrow A).functor.map
(((CostructuredArrow.toOverCompCoyoneda A).inv.app (op X)).app T f) =
(CostructuredArrow.toOverCompOverEquivPresheafCostructuredArrow A).hom.app X ≫ f :=
by simp [CostructuredArrow.toOverCompCoyoneda]
