-- Module: Mathlib.CategoryTheory.Limits.Presheaf
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)
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))))
eq₂ :
  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 g)).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))))
h :
  Eq (α :=
    Functor.obj uliftYoneda.{max v₂ w, v₁, u₁} (unop ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c))) ⟶
      F.op ⋙ (StructuredArrow.right Φ).obj P)
    ((Equiv.symm uliftYonedaEquiv :
            (a : P.obj (unop x).fst) → uliftYoneda.{max v₂ w, v₁, u₁}.obj (unop (unop x).fst) ⟶ P)
          ((unop x).snd (β := fun (c : Cᵒᵖ) ↦ P.obj c)) ≫
        NatTrans.app F.op.lanUnit P ≫ F.op.whiskerLeft ((StructuredArrow.Hom.right f).app P) :
      Functor.obj uliftYoneda.{max v₂ w, v₁, u₁} (unop ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c))) ⟶
        F.op ⋙ (StructuredArrow.right Φ).obj P)
    ((Equiv.symm uliftYonedaEquiv :
            (a : P.obj (unop x).fst) → uliftYoneda.{max v₂ w, v₁, u₁}.obj (unop (unop x).fst) ⟶ P)
          ((unop x).snd (β := fun (c : Cᵒᵖ) ↦ P.obj c)) ≫
        NatTrans.app F.op.lanUnit P ≫ F.op.whiskerLeft ((StructuredArrow.Hom.right g).app P) :
      Functor.obj uliftYoneda.{max v₂ w, v₁, u₁} (unop ((unop x).fst (β := fun (c : Cᵒᵖ) ↦ P.obj c))) ⟶
        F.op ⋙ (StructuredArrow.right Φ).obj 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)

Tactic:
dsimp at eq₁ eq₂ eq ⊢