-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Presheaf (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : ∀ (P : Cᵒᵖ ⥤ Type (max w v₁ v₂)), F.op.HasLeftKanExtension P
Φ : Functor.LeftExtension uliftYoneda.{max w v₂, v₁, u₁} (F ⋙ uliftYoneda.{max w v₁, v₂, u₂})
f g : Functor.LeftExtension.mk F.op.lan (Iso.hom (compULiftYonedaIsoULiftYonedaCompLan.{w, v₁, v₂, u₁, u₂} F)) ⟶ Φ
P : Cᵒᵖ ⥤ Type (max (max w v₁) v₂)
x : P.Elementsᵒᵖ
eq :
  Eq (α :=
    Functor.obj (𝟭 (Cᵒᵖ ⥤ Type (max (max w v₂) v₁))) (Functor.obj uliftYoneda.{max w v₂, v₁, u₁} (unop (unop x).1)) ⟶
      Functor.obj (F.op.lan ⋙ Functor.obj (Functor.whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max (max w v₂) v₁))) F.op) P)
    (Functor.map (𝟭 (Cᵒᵖ ⥤ Type (max (max w v₂) v₁)))
          ((Equiv.symm uliftYonedaEquiv :
              (a : P.obj (op (unop (unop x).1))) → uliftYoneda.{max w v₂, v₁, u₁}.obj (unop (unop x).1) ⟶ P)
            ((unop x).snd (β := fun (c : Cᵒᵖ) ↦ P.obj c))) ≫
        NatTrans.app F.op.lanUnit P :
      Functor.obj (𝟭 (Cᵒᵖ ⥤ Type (max (max w v₂) v₁))) (Functor.obj uliftYoneda.{max w v₂, v₁, u₁} (unop (unop x).1)) ⟶
        Functor.obj (F.op.lan ⋙ Functor.obj (Functor.whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max (max w v₂) v₁))) F.op) P)
    (NatTrans.app F.op.lanUnit (Functor.obj uliftYoneda.{max w v₂, v₁, u₁} (unop (unop x).1)) ≫
        Functor.map (F.op.lan ⋙ Functor.obj (Functor.whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max (max w v₂) v₁))) F.op)
          ((Equiv.symm uliftYonedaEquiv :
              (a : P.obj (op (unop (unop x).1))) → uliftYoneda.{max w v₂, v₁, u₁}.obj (unop (unop x).1) ⟶ P)
            ((unop x).snd (β := fun (c : Cᵒᵖ) ↦ P.obj c))) :
      Functor.obj (𝟭 (Cᵒᵖ ⥤ Type (max (max w v₂) v₁))) (Functor.obj uliftYoneda.{max w v₂, v₁, u₁} (unop (unop x).1)) ⟶
        Functor.obj (F.op.lan ⋙ Functor.obj (Functor.whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max (max w v₂) v₁))) F.op) P)
⊢ (coconeOfRepresentable P).ι.app x ≫
      NatTrans.app F.op.lanUnit P ≫ F.op.whiskerLeft ((StructuredArrow.Hom.right f).app P) =
    (coconeOfRepresentable P).ι.app x ≫
      NatTrans.app F.op.lanUnit P ≫ F.op.whiskerLeft ((StructuredArrow.Hom.right g).app P)
have eq₁ : Eq (α :=
  ((Functor.obj
            (Functor.obj (Functor.whiskeringLeft C (Cᵒᵖ ⥤ Type (max (max w v₁) v₂)) (Dᵒᵖ ⥤ Type (max (max w v₁) v₂)))
              uliftYoneda.{max w v₂, v₁, u₁})
            (StructuredArrow.right Φ)).obj
        (unop ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c)))).obj
    (F.op.obj ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c))))
  ((hom (FC := TypeCat.Fun)
          (NatTrans.app
            ((StructuredArrow.hom
                    (Functor.LeftExtension.mk F.op.lan
                      (Iso.hom (compULiftYonedaIsoULiftYonedaCompLan.{w, v₁, v₂, u₁, u₂} F))) ≫
                  Functor.map
                    (Functor.obj
                      (Functor.whiskeringLeft C (Cᵒᵖ ⥤ Type (max (max w v₁) v₂)) (Dᵒᵖ ⥤ Type (max (max w v₁) v₂)))
                      uliftYoneda.{max w v₂, v₁, u₁})
                    (StructuredArrow.Hom.right f)).app
              (unop ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c))))
            (F.op.obj ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c)))) :
        (a : ((F ⋙ uliftYoneda.{max w v₁, v₂, u₂}).obj (unop (unop x).fst)).obj (F.op.obj (unop x).fst)) →
          ((((Functor.whiskeringLeft C (Cᵒᵖ ⥤ Type (max (max w v₁) v₂)) (Dᵒᵖ ⥤ Type (max (max w v₁) v₂))).obj
                        uliftYoneda.{max w v₂, v₁, u₁}).obj
                    (StructuredArrow.right Φ)).obj
                (unop (unop x).fst)).obj
            (F.op.obj (unop x).fst))
      { down := 𝟙 (unop (F.op.obj ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c)))) } :
    ((Functor.obj
              (Functor.obj (Functor.whiskeringLeft C (Cᵒᵖ ⥤ Type (max (max w v₁) v₂)) (Dᵒᵖ ⥤ Type (max (max w v₁) v₂)))
                uliftYoneda.{max w v₂, v₁, u₁})
              (StructuredArrow.right Φ)).obj
          (unop ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c)))).obj
      (F.op.obj ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c))))
  ((hom (FC := TypeCat.Fun)
          (NatTrans.app ((StructuredArrow.hom Φ).app (unop ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c))))
            (F.op.obj ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c)))) :
        (a : ((F ⋙ uliftYoneda.{max w v₁, v₂, u₂}).obj (unop (unop x).fst)).obj (F.op.obj (unop x).fst)) →
          ((((Functor.whiskeringLeft C (Cᵒᵖ ⥤ Type (max (max w v₁) v₂)) (Dᵒᵖ ⥤ Type (max (max w v₁) v₂))).obj
                        uliftYoneda.{max w v₂, v₁, u₁}).obj
                    (StructuredArrow.right Φ)).obj
                (unop (unop x).fst)).obj
            (F.op.obj (unop x).fst))
      { down := 𝟙 (unop (F.op.obj ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c)))) } :
    ((Functor.obj
              (Functor.obj (Functor.whiskeringLeft C (Cᵒᵖ ⥤ Type (max (max w v₁) v₂)) (Dᵒᵖ ⥤ Type (max (max w v₁) v₂)))
                uliftYoneda.{max w v₂, v₁, u₁})
              (StructuredArrow.right Φ)).obj
          (unop ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c)))).obj
      (F.op.obj ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c))))
Tactic: have eq₁ := congr_hom (CC := fun X ↦ X) (congr_app (congr_app (StructuredArrow.w f) x.unop.1.unop)
    (F.op.obj x.unop.1)) (ULift.up (𝟙 _))