-- Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
/-- The square `precomposeTransformSquare` respects compositions.  -/
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjTransformObjSquare_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}
  {X : Type u₇} {Y : Type u₈} {Z : Type u₉} [inst_6 : CategoryTheory.Category.{v₇, u₇} X]
  [inst_7 : CategoryTheory.Category.{v₈, u₈} Y] [inst_8 : CategoryTheory.Category.{v₉, u₉} Z]
  (ψ : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁) (U : CategoryTheory.Functor X Y)
  (V : CategoryTheory.Functor Y Z),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.CatCommSq.iso
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj (U.comp V))
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Z).obj ψ)
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ)
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj (U.comp V))).hom
      (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Z).obj ψ).whiskerLeft
        (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp F₁ G₁ U V).hom) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Functor.whiskerRight
        (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp F G U V).hom
        ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ))
      (CategoryTheory.CategoryStruct.comp
        (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V).associator
            ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U)
            ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ)).hom
        (CategoryTheory.CategoryStruct.comp
          (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V).whiskerLeft
            (CategoryTheory.CatCommSq.iso
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U)
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ)
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ)
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)).hom)
          (CategoryTheory.CategoryStruct.comp
            (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V).associator
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ)
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)).inv
            (CategoryTheory.CategoryStruct.comp
              (CategoryTheory.Functor.whiskerRight
                (CategoryTheory.CatCommSq.iso
                    ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V)
                    ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Z).obj ψ)
                    ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ)
                    ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj V)).hom
                ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U))
              (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Z).obj ψ).associator
                  ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj V)
                  ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)).hom))))