Theorems · Theorem · category theory

CategoryTheory.presheafHom_map_app

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {A : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} A]
  {F G : CategoryTheory.Functor Cᵒᵖ A} {X Y Z : C} (f : Z ⟶ Y) (g : Y ⟶ X) (h : Z ⟶ X),
  CategoryTheory.CategoryStruct.comp f g = h →
    ∀ (α : (CategoryTheory.presheafHom F G).obj (Opposite.op X)),
      ((CategoryTheory.ConcreteCategory.hom ((CategoryTheory.presheafHom F G).map g.op)) α).app
          (Opposite.op (CategoryTheory.Over.mk f)) =
        α.app (Opposite.op (CategoryTheory.Over.mk h))

Equational lemma for the presheaf structure on presheafHom. It is advisable to use this lemma rather than dsimp [presheafHom] which may result in the need to prove equalities of objects in an Over category.

Defined in

Mathlib.CategoryTheory.Sites.SheafHom

Cited by

2 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Category