-- Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument
Goal:
C : Type u
inst : Category.{v, u} C
I : MorphismProperty.{v, u} C
κ : Cardinal.{w}
inst_1 : Fact κ.IsRegular
inst_2 : OrderBot κ.ord.ToType
inst_3 : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this_1 : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this_2 : HasPushouts.{v, u} C
this :
  PreservesColimit
    (TransfiniteCompositionOfShape.F (Y := obj I κ f)
      (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦ (↑I.toSet : Type (max u v)))
        (relativeCellComplexιObj I κ f)))
    (Functor.obj coyoneda (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t :
  A ⟶
    (TransfiniteCompositionOfShape.F (Y := obj I κ f)
          (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
            (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f))).obj
      j
ht :
  t ≫
      NatTrans.app
        (TransfiniteCompositionOfShape.incl
          (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
            (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f)))
        j =
    g
rw :
  (TransfiniteCompositionOfShape.F (Y := obj I κ f)
            (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
              (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f))).map
        (homOfLE ⋯) ≫
      Iso.hom (relativeCellComplexιObjFObjSuccIso I κ f j) =
    𝟙
        ((TransfiniteCompositionOfShape.F (Y := obj I κ f)
              (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
                (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f))).obj
          j) ≫
      (TransfiniteCompositionOfShape.F (Y := obj I κ f)
              (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
                (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f))).map
          (homOfLE ⋯) ≫
        Iso.hom (relativeCellComplexιObjFObjSuccIso I κ f j)
rw₁ rw₂ :
  (TransfiniteCompositionOfShape.F (Y := obj I κ f)
            (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
              (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f))).map
        (homOfLE ⋯) ≫
      Iso.hom (relativeCellComplexιObjFObjSuccIso I κ f j) =
    (TransfiniteCompositionOfShape.F (Y := obj I κ f)
            (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
              (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f))).map
        (homOfLE ⋯) ≫
      Iso.hom (relativeCellComplexιObjFObjSuccIso I κ f j)
⊢ (TransfiniteCompositionOfShape.F (Y := obj I κ f)
            (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
              (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f))).map
        (homOfLE ⋯) ≫
      Iso.hom (relativeCellComplexιObjFObjSuccIso I κ f j) =
    𝟙
        ((TransfiniteCompositionOfShape.F (Y := obj I κ f)
              (RelativeCellComplex.toTransfiniteCompositionOfShape (α := fun (x : κ.ord.ToType) ↦
                (↑I.toSet : Type (max u v))) (relativeCellComplexιObj I κ f))).obj
          j) ≫
      ιFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom

Tactic:
rw [ιFunctorObj_eq, id_comp]

Premises used: CategoryTheory.SmallObject.ιFunctorObj_eq, CategoryTheory.Category.id_comp