Theorems · Definition · category theory

CategoryTheory.Functor.mapCoconePrecompose

{J : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} J] →
    {C : Type u₃} →
      [inst_1 : CategoryTheory.Category.{v₃, u₃} C] →
        {D : Type u₄} →
          [inst_2 : CategoryTheory.Category.{v₄, u₄} D] →
            (H : CategoryTheory.Functor C D) →
              {F G : CategoryTheory.Functor J C} →
                {α : G ⟶ F} →
                  {c : CategoryTheory.Limits.Cocone F} →
                    H.mapCocone ((CategoryTheory.Limits.Cocone.precompose α).obj c) ≅
                      (CategoryTheory.Limits.Cocone.precompose (CategoryTheory.Functor.whiskerRight α H)).obj
                        (H.mapCocone c)

map_cocone commutes with precompose. In particular, for F : J ⥤ C, given a cocone c : Cocone F, a natural transformation α : F ⟶ G and a functor H : C ⥤ D, we have two obvious ways of producing a cocone over G ⋙ H, and they are both isomorphic.

Defined in

Mathlib.CategoryTheory.Limits.Cones

Cited by

2 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Category