Documentation

Mathlib.CategoryTheory.Functor.Hom

The hom functor, sending (X, Y) to the type X ⟶ Y.

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def CategoryTheory.Functor.hom (C : Type u) [CategoryTheory.Category.{v, u} C] :
CategoryTheory.Functor (Cᵒᵖ × C) (Type v)

Functor.hom is the hom-pairing, sending (X, Y) to X ⟶ Y, contravariant in X and covariant in Y.

Equations
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Instances For
    source
    @[simp]
    theorem CategoryTheory.Functor.hom_obj (C : Type u) [CategoryTheory.Category.{v, u} C] (p : Cᵒᵖ × C) :
    (CategoryTheory.Functor.hom C).obj p = (Opposite.unop p.1 ⟶ p.2)
    source
    @[simp]
    theorem CategoryTheory.Functor.hom_map (C : Type u) [CategoryTheory.Category.{v, u} C] :
    ∀ {X Y : Cᵒᵖ × C} (f : X ⟶ Y) (h : Opposite.unop X.1 ⟶ X.2), (CategoryTheory.Functor.hom C).map f h = CategoryTheory.CategoryStruct.comp f.1.unop (CategoryTheory.CategoryStruct.comp h f.2)