Theorems · Definition · category theory

CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp

{A : Type u₁} →
  {B : Type u₂} →
    {C : Type u₃} →
      [inst : CategoryTheory.Category.{v₁, u₁} A] →
        [inst_1 : CategoryTheory.Category.{v₂, u₂} B] →
          [inst_2 : CategoryTheory.Category.{v₃, u₃} C] →
            (F : CategoryTheory.Functor A B) →
              (G : CategoryTheory.Functor C B) →
                {X : Type u₄} →
                  {Y : Type u₅} →
                    {Z : Type u₆} →
                      [inst_3 : CategoryTheory.Category.{v₄, u₄} X] →
                        [inst_4 : CategoryTheory.Category.{v₅, u₅} Y] →
                          [inst_5 : CategoryTheory.Category.{v₆, u₆} Z] →
                            (U : CategoryTheory.Functor X Y) →
                              (V : CategoryTheory.Functor Y Z) →
                                (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj
                                    (U.comp V) ≅
                                  ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V).comp
                                    ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U)

The construction precompose respects functor composition.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic

Cited by

10 results in Mathlib

Foundations

Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Category