Theorems · Theorem · category theory

CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjPrecomposeObjSquare_iso_hom_app_snd_app

∀ {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} {A₁ : Type u₄} {B₁ : Type u₅} {C₁ : Type u₆}
  [inst_3 : CategoryTheory.Category.{v₄, u₄} A₁] [inst_4 : CategoryTheory.Category.{v₅, u₅} B₁]
  [inst_5 : CategoryTheory.Category.{v₆, u₆} C₁] {F₁ : CategoryTheory.Functor A₁ B₁} {G₁ : CategoryTheory.Functor C₁ B₁}
  {X : Type u₇} {Y : Type u₈} [inst_6 : CategoryTheory.Category.{v₇, u₇} X]
  [inst_7 : CategoryTheory.Category.{v₈, u₈} Y] (U : CategoryTheory.Functor X Y)
  (ψ : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁)
  (X_1 : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (x : X),
  ((CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ)
                  ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U)
                  ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)
                  ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ)).hom.app
            X_1).snd.app
      x =
    CategoryTheory.CategoryStruct.id (ψ.right.obj (X_1.snd.obj (U.obj x)))

Defined in

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

Cited by

0 results in Mathlib

Foundations

Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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