-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic (hypothesis)
Goal:
A : Type u₁
B : Type u₂
C : Type u₃
inst : Category.{v₁, u₁} A
inst_1 : Category.{v₂, u₂} B
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₂, u₁, u₂} A B
G : Functor.{v₃, v₂, u₃, u₂} C B
A₁ : Type u₄
B₁ : Type u₅
C₁ : Type u₆
inst_3 : Category.{v₄, u₄} A₁
inst_4 : Category.{v₅, u₅} B₁
inst_5 : Category.{v₆, u₆} C₁
F₁ : Functor.{v₄, v₅, u₄, u₅} A₁ B₁
G₁ : Functor.{v₆, v₅, u₆, u₅} C₁ B₁
X : Type u₇
inst_6 : Category.{v₇, u₇} X
ψ ψ' : CatCospanTransform F G F₁ G₁
η : ψ ⟶ ψ'
S : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X
t : X
⊢ Eq (α :=
    obj.{v₇, v₅, u₇, u₅}
        (fst.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} (B := B₁) (C := C₁) (F := F₁) (G := G₁)
            (obj
              {
                obj := fun (S : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X) ↦
                  { fst := S.fst ⋙ ψ.left, snd := S.snd ⋙ ψ.right,
                    iso :=
                      S.fst.associator ψ.left F₁ ≪≫
                        S.fst.isoWhiskerLeft (CatCommSq.iso F ψ.left ψ.base F₁).symm ≪≫
                          (S.fst.associator F ψ.base).symm ≪≫
                            isoWhiskerRight S.iso ψ.base ≪≫
                              S.snd.associator G ψ.base ≪≫
                                S.snd.isoWhiskerLeft (CatCommSq.iso G ψ.right ψ.base G₁) ≪≫
                                  (S.snd.associator ψ.right G₁).symm :
                    CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X },
                map := fun {x y : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X} (f : x ⟶ y) ↦
                  { fst := whiskerRight f.fst ψ.left, snd := whiskerRight f.snd ψ.right, w := ⋯ :
                    Hom X
                      { fst := x.fst ⋙ ψ.left, snd := x.snd ⋙ ψ.right,
                        iso :=
                          x.fst.associator ψ.left F₁ ≪≫
                            x.fst.isoWhiskerLeft (CatCommSq.iso F ψ.left ψ.base F₁).symm ≪≫
                              (x.fst.associator F ψ.base).symm ≪≫
                                isoWhiskerRight x.iso ψ.base ≪≫
                                  x.snd.associator G ψ.base ≪≫
                                    x.snd.isoWhiskerLeft (CatCommSq.iso G ψ.right ψ.base G₁) ≪≫
                                      (x.snd.associator ψ.right G₁).symm :
                        CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X }
                      { fst := y.fst ⋙ ψ.left, snd := y.snd ⋙ ψ.right,
                        iso :=
                          y.fst.associator ψ.left F₁ ≪≫
                            y.fst.isoWhiskerLeft (CatCommSq.iso F ψ.left ψ.base F₁).symm ≪≫
                              (y.fst.associator F ψ.base).symm ≪≫
                                isoWhiskerRight y.iso ψ.base ≪≫
                                  y.snd.associator G ψ.base ≪≫
                                    y.snd.isoWhiskerLeft (CatCommSq.iso G ψ.right ψ.base G₁) ≪≫
                                      (y.snd.associator ψ.right G₁).symm :
                        CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X } },
                map_id := ⋯, map_comp := ⋯ :
                CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X ⥤
                  CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X }
              S) ⋙
          F₁)
        t ⟶
      obj.{v₇, v₅, u₇, u₅}
        (snd.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} (A := A₁) (B := B₁) (F := F₁) (G := G₁)
            (obj
              {
                obj := fun (S : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X) ↦
                  { fst := S.fst ⋙ ψ'.left, snd := S.snd ⋙ ψ'.right,
                    iso :=
                      S.fst.associator ψ'.left F₁ ≪≫
                        S.fst.isoWhiskerLeft (CatCommSq.iso F ψ'.left ψ'.base F₁).symm ≪≫
                          (S.fst.associator F ψ'.base).symm ≪≫
                            isoWhiskerRight S.iso ψ'.base ≪≫
                              S.snd.associator G ψ'.base ≪≫
                                S.snd.isoWhiskerLeft (CatCommSq.iso G ψ'.right ψ'.base G₁) ≪≫
                                  (S.snd.associator ψ'.right G₁).symm },
                map := fun {x y : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X} (f : x ⟶ y) ↦
                  { fst := whiskerRight f.fst ψ'.left, snd := whiskerRight f.snd ψ'.right, w := ⋯ },
                map_id := ⋯, map_comp := ⋯ }
              S) ⋙
          G₁)
        t)
    (NatTrans.app
      (whiskerRight { app := fun (y : X) ↦ η.left.app (S.fst.obj y), naturality := ⋯ } F₁ ≫
        Iso.hom
          (iso
            (obj
              {
                obj := fun (S : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X) ↦
                  { fst := S.fst ⋙ ψ'.left, snd := S.snd ⋙ ψ'.right,
                    iso :=
                      S.fst.associator ψ'.left F₁ ≪≫
                        S.fst.isoWhiskerLeft (CatCommSq.iso F ψ'.left ψ'.base F₁).symm ≪≫
                          (S.fst.associator F ψ'.base).symm ≪≫
                            isoWhiskerRight S.iso ψ'.base ≪≫
                              S.snd.associator G ψ'.base ≪≫
                                S.snd.isoWhiskerLeft (CatCommSq.iso G ψ'.right ψ'.base G₁) ≪≫
                                  (S.snd.associator ψ'.right G₁).symm :
                    CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X },
                map := fun {x y : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X} (f : x ⟶ y) ↦
                  { fst := whiskerRight f.fst ψ'.left, snd := whiskerRight f.snd ψ'.right, w := ⋯ :
                    Hom X
                      { fst := x.fst ⋙ ψ'.left, snd := x.snd ⋙ ψ'.right,
                        iso :=
                          x.fst.associator ψ'.left F₁ ≪≫
                            x.fst.isoWhiskerLeft (CatCommSq.iso F ψ'.left ψ'.base F₁).symm ≪≫
                              (x.fst.associator F ψ'.base).symm ≪≫
                                isoWhiskerRight x.iso ψ'.base ≪≫
                                  x.snd.associator G ψ'.base ≪≫
                                    x.snd.isoWhiskerLeft (CatCommSq.iso G ψ'.right ψ'.base G₁) ≪≫
                                      (x.snd.associator ψ'.right G₁).symm :
                        CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X }
                      { fst := y.fst ⋙ ψ'.left, snd := y.snd ⋙ ψ'.right,
                        iso :=
                          y.fst.associator ψ'.left F₁ ≪≫
                            y.fst.isoWhiskerLeft (CatCommSq.iso F ψ'.left ψ'.base F₁).symm ≪≫
                              (y.fst.associator F ψ'.base).symm ≪≫
                                isoWhiskerRight y.iso ψ'.base ≪≫
                                  y.snd.associator G ψ'.base ≪≫
                                    y.snd.isoWhiskerLeft (CatCommSq.iso G ψ'.right ψ'.base G₁) ≪≫
                                      (y.snd.associator ψ'.right G₁).symm :
                        CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X } },
                map_id := ⋯, map_comp := ⋯ :
                CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X ⥤
                  CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X }
              S)))
      t)
    (NatTrans.app
      (Iso.hom
          (iso
            (obj
              {
                obj := fun (S : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X) ↦
                  { fst := S.fst ⋙ ψ.left, snd := S.snd ⋙ ψ.right,
                    iso :=
                      S.fst.associator ψ.left F₁ ≪≫
                        S.fst.isoWhiskerLeft (CatCommSq.iso F ψ.left ψ.base F₁).symm ≪≫
                          (S.fst.associator F ψ.base).symm ≪≫
                            isoWhiskerRight S.iso ψ.base ≪≫
                              S.snd.associator G ψ.base ≪≫
                                S.snd.isoWhiskerLeft (CatCommSq.iso G ψ.right ψ.base G₁) ≪≫
                                  (S.snd.associator ψ.right G₁).symm :
                    CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X },
                map := fun {x y : CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X} (f : x ⟶ y) ↦
                  { fst := whiskerRight f.fst ψ.left, snd := whiskerRight f.snd ψ.right, w := ⋯ :
                    Hom X
                      { fst := x.fst ⋙ ψ.left, snd := x.snd ⋙ ψ.right,
                        iso :=
                          x.fst.associator ψ.left F₁ ≪≫
                            x.fst.isoWhiskerLeft (CatCommSq.iso F ψ.left ψ.base F₁).symm ≪≫
                              (x.fst.associator F ψ.base).symm ≪≫
                                isoWhiskerRight x.iso ψ.base ≪≫
                                  x.snd.associator G ψ.base ≪≫
                                    x.snd.isoWhiskerLeft (CatCommSq.iso G ψ.right ψ.base G₁) ≪≫
                                      (x.snd.associator ψ.right G₁).symm :
                        CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X }
                      { fst := y.fst ⋙ ψ.left, snd := y.snd ⋙ ψ.right,
                        iso :=
                          y.fst.associator ψ.left F₁ ≪≫
                            y.fst.isoWhiskerLeft (CatCommSq.iso F ψ.left ψ.base F₁).symm ≪≫
                              (y.fst.associator F ψ.base).symm ≪≫
                                isoWhiskerRight y.iso ψ.base ≪≫
                                  y.snd.associator G ψ.base ≪≫
                                    y.snd.isoWhiskerLeft (CatCommSq.iso G ψ.right ψ.base G₁) ≪≫
                                      (y.snd.associator ψ.right G₁).symm :
                        CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X } },
                map_id := ⋯, map_comp := ⋯ :
                CatCommSqOver.{v₁, v₂, v₃, v₇, u₁, u₂, u₃, u₇} F G X ⥤
                  CatCommSqOver.{v₄, v₅, v₆, v₇, u₄, u₅, u₆, u₇} F₁ G₁ X }
              S)) ≫
        whiskerRight { app := fun (y : X) ↦ η.right.app (S.snd.obj y), naturality := ⋯ } G₁)
      t)
have this : Eq (α := (ψ.left ⋙ F₁).obj (S.fst.obj t) ⟶ F₁.obj (ψ'.left.obj (S.fst.obj t)))
  ((CatCommSq.iso F ψ.left ψ.base F₁).inv.app (S.fst.obj t) ≫
    (CatCommSq.iso F ψ.left ψ.base F₁).hom.app (S.fst.obj t) ≫ F₁.map (η.left.app (S.fst.obj t)))
  ((CatCommSq.iso F ψ.left ψ.base F₁).inv.app (S.fst.obj t) ≫
    η.base.app (F.obj (S.fst.obj t)) ≫ (CatCommSq.iso F ψ'.left ψ'.base F₁).hom.app (S.fst.obj t))
Tactic: have := ψ.squareLeft.iso.inv.app (S.fst.obj t) ≫=
            η.left_coherence_app (S.fst.obj t)