-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C D : Type u₁
inst : SmallCategory C
inst_1 : SmallCategory D
E : Type u₂
inst_2 : Category.{u₁, u₂} E
F : C ⥤ D
X : D
inst_3 : ∀ (X : D), HasColimitsOfShape.{u₁, u₁, u₁, u₂} (CostructuredArrow F X) E
G₁ G₂ : C ⥤ E
φ : G₁ ⟶ G₂
T : CostructuredArrow F X
h₁ :
  ∀ (G : C ⥤ E),
    Eq (α := G.obj T.left ⟶ colimit (CostructuredArrow.proj F X ⋙ G))
      ((NatTrans.app (F.leftKanExtensionUnit G) T.left ≫ (F.leftKanExtension G).map T.hom) ≫
        Iso.hom
          (IsColimit.coconePointUniqueUpToIso (F.isPointwiseLeftKanExtensionLeftKanExtensionUnit G X)
            (colimit.isColimit (CostructuredArrow.proj F X ⋙ G))))
      (colimit.ι (CostructuredArrow.proj F X ⋙ G) T)
h₂ :
  Eq (α := G₁.obj T.left ⟶ (Functor.obj F.lan G₂).obj (F.obj T.left))
    (φ.app T.left ≫ NatTrans.app (NatTrans.app F.lanUnit G₂) T.left)
    (NatTrans.app (NatTrans.app F.lanUnit G₁) T.left ≫ NatTrans.app (Functor.map F.lan φ) (F.obj T.left))
simp :
  ∀
    (h₁ :
      ∀ (G : C ⥤ E),
        Eq (α := G.obj T.left ⟶ colimit (CostructuredArrow.proj F X ⋙ G))
          (NatTrans.app (F.leftKanExtensionUnit G) T.left ≫
            (F.leftKanExtension G).map T.hom ≫
              Iso.hom
                (IsColimit.coconePointUniqueUpToIso (F.isPointwiseLeftKanExtensionLeftKanExtensionUnit G X)
                  (colimit.isColimit (CostructuredArrow.proj F X ⋙ G))))
          (colimit.ι (CostructuredArrow.proj F X ⋙ G) T)),
    Eq (α := G₁.obj T.left ⟶ colimit (CostructuredArrow.proj F X ⋙ G₂))
      (NatTrans.app (F.leftKanExtensionUnit G₁) T.left ≫
        (F.leftKanExtension G₁).map T.hom ≫
          NatTrans.app (Functor.map F.lan φ) X ≫
            Iso.hom
              (IsColimit.coconePointUniqueUpToIso (F.isPointwiseLeftKanExtensionLeftKanExtensionUnit G₂ X)
                (colimit.isColimit (CostructuredArrow.proj F X ⋙ G₂))))
      (NatTrans.app (F.leftKanExtensionUnit G₁) T.left ≫
        (F.leftKanExtension G₁).map T.hom ≫
          CategoryStruct.comp (Y := colimit (CostructuredArrow.proj F X ⋙ G₁))
            (Iso.hom
              (IsColimit.coconePointUniqueUpToIso (F.isPointwiseLeftKanExtensionLeftKanExtensionUnit G₁ X)
                (colimit.isColimit (CostructuredArrow.proj F X ⋙ G₁))))
            (colimMap ((CostructuredArrow.proj F X).whiskerLeft φ)))
⊢ Eq (α := G₁.obj T.left ⟶ colimit (CostructuredArrow.proj F X ⋙ G₂))
    ((NatTrans.app (F.leftKanExtensionUnit G₁) T.left ≫ (F.leftKanExtension G₁).map T.hom) ≫
      NatTrans.app (Functor.map F.lan φ) X ≫
        Iso.hom
          (IsColimit.coconePointUniqueUpToIso (F.isPointwiseLeftKanExtensionLeftKanExtensionUnit G₂ X)
            (colimit.isColimit (CostructuredArrow.proj F X ⋙ G₂))))
    ((NatTrans.app (F.leftKanExtensionUnit G₁) T.left ≫ (F.leftKanExtension G₁).map T.hom) ≫
      CategoryStruct.comp (Y := colimit (CostructuredArrow.proj F X ⋙ G₁))
        (Iso.hom
          (IsColimit.coconePointUniqueUpToIso (F.isPointwiseLeftKanExtensionLeftKanExtensionUnit G₁ X)
            (colimit.isColimit (CostructuredArrow.proj F X ⋙ G₁))))
        (colimMap ((CostructuredArrow.proj F X).whiskerLeft φ)))

Tactic:
simp only [Category.assoc] at h₁ ⊢

Premises used: CategoryTheory.Category.assoc