Theorems · Theorem · category theory

CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformPrecomposeObjSquare_iso_hom_comp

∀ {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}
  {A₂ : Type u₇} {B₂ : Type u₈} {C₂ : Type u₉} [inst_6 : CategoryTheory.Category.{v₇, u₇} A₂]
  [inst_7 : CategoryTheory.Category.{v₈, u₈} B₂] [inst_8 : CategoryTheory.Category.{v₉, u₉} C₂]
  {F₂ : CategoryTheory.Functor A₂ B₂} {G₂ : CategoryTheory.Functor C₂ B₂} {X : Type u₁₀} {Y : Type u₁₁}
  [inst_9 : CategoryTheory.Category.{v₁₀, u₁₀} X] [inst_10 : CategoryTheory.Category.{v₁₁, u₁₁} Y]
  (U : CategoryTheory.Functor X Y) (ψ : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁)
  (ψ' : CategoryTheory.Limits.CatCospanTransform F₁ G₁ F₂ G₂),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.CatCommSq.iso
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj (ψ.comp ψ'))
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U)
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₂ G₂).obj U)
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj (ψ.comp ψ'))).hom
      (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U).whiskerLeft
        (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp X ψ ψ').hom) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Functor.whiskerRight
        (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp Y ψ ψ').hom
        ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₂ G₂).obj U))
      (CategoryTheory.CategoryStruct.comp
        (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ).associator
            ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ')
            ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₂ G₂).obj U)).hom
        (CategoryTheory.CategoryStruct.comp
          (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ).whiskerLeft
            (CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ')
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₂ G₂).obj U)
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ')).hom)
          (CategoryTheory.CategoryStruct.comp
            (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ).associator
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ')).inv
            (CategoryTheory.CategoryStruct.comp
              (CategoryTheory.Functor.whiskerRight
                (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
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ'))
              (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U).associator
                  ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ)
                  ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ')).hom))))

The square transformPrecomposeSquare respects compositions.

Defined in

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

Cited by

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Foundations

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

Assumes

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