English
In the costructured-arrow setting, the inverse map of toOverCompYoneda acts as described by the stated equality involving isoCompInverse and unit.
Русский
В контексте CostructuredArrow, образованный обратный отображение для toOverCompYoneda действует согласно указанному равенству с isoCompInverse и единицей.
LaTeX
$$$$ (overEquivPresheafCostructuredArrow A)^{-1}.map\ (((CostructuredArrow.toOverCompYoneda A).hom.app { op X }).app T f) \,=\ (CostructuredArrow.toOverCompOverEquivPresheafCostructuredArrow A).isoCompInverse.inv.app X \\gg f \\gg (overEquivPresheafCostructuredArrow A).unit.app T $$$$
Lean4
@[simp]
theorem overEquivPresheafCostructuredArrow_inverse_map_toOverCompYoneda {A : Cᵒᵖ ⥤ Type v} {T : Over A}
{X : CostructuredArrow yoneda A} (f : (CostructuredArrow.toOver yoneda A).obj X ⟶ T) :
(overEquivPresheafCostructuredArrow A).inverse.map (((CostructuredArrow.toOverCompYoneda A T).hom.app (op X) f)) =
(CostructuredArrow.toOverCompOverEquivPresheafCostructuredArrow A).isoCompInverse.inv.app X ≫
f ≫ (overEquivPresheafCostructuredArrow A).unit.app T :=
by simp [CostructuredArrow.toOverCompYoneda]
