theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
    {F : J ⥤ D} {c : Cocone (F ⋙ Gr)} (H : IsVanKampenColimit c)
    [∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
    [∀ X (f : X ⟶ Gl.obj c.pt), PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl]
    [∀ X i (f : X ⟶ c.pt), PreservesLimit (cospan f (c.ι.app i)) Gl] :
    IsVanKampenColimit (Gl.mapCocone c) := by
  have := adj.rightAdjointPreservesLimits
  have : PreservesColimitsOfSize.{u', v'} Gl := adj.leftAdjointPreservesColimits
  intro F' c' α f h hα
  refine ⟨?_, H.isUniversal.map_reflective adj c' α f h hα⟩
  intro ⟨hc'⟩ j
  let α' := α ≫ (Functor.associator _ _ _).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
  have hα' : NatTrans.Equifibered α' := hα.comp (NatTrans.equifibered_of_isIso _)
  have hα'' : ∀ j, Gl.map (Gr.map $ α'.app j) = adj.counit.app _ ≫ α.app j
  · intro j
    rw [← cancel_mono (adj.counit.app $ F.obj j)]
    dsimp
    simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
      Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
  let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
  let hl := (IsColimit.precomposeHomEquiv β c').symm hc'
  let hr := isColimitOfPreserves Gl (colimit.isColimit $ F' ⋙ Gr)
  have : α.app j = β.inv.app _ ≫ Gl.map (Gr.map $ α'.app j)
  · rw [hα'']
    simp
  rw [this]
  have : f = (hl.coconePointUniqueUpToIso hr).hom ≫
    Gl.map (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩)
  · symm
    convert @IsColimit.coconePointUniqueUpToIso_hom_desc _ _ _ _ ((F' ⋙ Gr) ⋙ Gl)
      (Gl.mapCocone ⟨_, (whiskerRight α' Gr ≫ c.2 : _)⟩) _ _ hl hr using 2
    · apply hr.hom_ext
      intro j
      rw [hr.fac, Functor.mapCocone_ι_app, ← Gl.map_comp, colimit.cocone_ι, colimit.ι_desc]
      rfl
    · clear_value α'
      apply hl.hom_ext
      intro j
      rw [hl.fac]
      dsimp
      simp only [Category.comp_id, hα'', Category.assoc, Gl.map_comp]
      congr 1
      exact (NatTrans.congr_app h j).symm
  rw [this]
  have := ((H (colimit.cocone $ F' ⋙ Gr) (whiskerRight α' Gr)
    (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩) ?_ (hα'.whiskerRight Gr)).mp
    ⟨(getColimitCocone $ F' ⋙ Gr).2⟩ j).map Gl
  convert IsPullback.paste_vert _ this
  refine IsPullback.of_vert_isIso ⟨?_⟩
  

Proof state:
case refine_2.convert_6
J : Type v'
inst✝⁹ : Category.{u', v'} J
C : Type u
inst✝⁸ : Category.{v, u} C
K : Type u_1
inst✝⁷ : Category.{?u.215693, u_1} K
D : Type u_2
inst✝⁶ : Category.{u_3, u_2} D
inst✝⁵ : HasColimitsOfShape J C
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝⁴ : Full Gr
inst✝³ : Faithful Gr
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsVanKampenColimit c
inst✝² : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)
inst✝¹ : (X : D) → (f : X ⟶ Gl.obj c.pt) → PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl
inst✝ : (X : C) → (i : J) → (f : X ⟶ c.pt) → PreservesLimit (cospan f (c.ι.app i)) Gl
this✝³ : PreservesLimitsOfSize.{0, 0, u_3, v, u_2, u} Gr
this✝² : PreservesColimitsOfSize.{u', v', v, u_3, u, u_2} Gl
F' : J ⥤ D
c' : Cocone F'
α : F' ⟶ (F ⋙ Gr) ⋙ Gl
f : c'.pt ⟶ (Gl.mapCocone c).pt
h : α ≫ (Gl.mapCocone c).ι = c'.ι ≫ (Functor.const J).map f
hα : NatTrans.Equifibered α
hc' : IsColimit c'
j : J
α' : F' ⟶ F := α ≫ (Functor.associator F Gr Gl).hom ≫ whiskerLeft F adj.counit ≫ (Functor.rightUnitor F).hom
hα' : NatTrans.Equifibered α'
hα'' : ∀ (j : J), Gl.map (Gr.map (α'.app j)) = adj.counit.app (F'.toPrefunctor.1 j) ≫ α.app j
β : F' ⋙ Gr ⋙ Gl ≅ F' := isoWhiskerLeft F' (asIso adj.counit) ≪≫ Functor.rightUnitor F'
hl : IsColimit ((Cocones.precompose β.hom).obj c') := (IsColimit.precomposeHomEquiv β c').symm hc'
hr : IsColimit (Gl.mapCocone (colimit.cocone (F' ⋙ Gr))) := isColimitOfPreserves Gl (colimit.isColimit (F' ⋙ Gr))
this✝¹ : α.app j = β.inv.app j ≫ Gl.map (Gr.map (α'.app j))
this✝ :
  f =
    (IsColimit.coconePointUniqueUpToIso hl hr).hom ≫
      Gl.map (colimit.desc (F' ⋙ Gr) { pt := c.pt, ι := whiskerRight α' Gr ≫ c.ι })
this :
  IsPullback (Gl.map ((colimit.cocone (F' ⋙ Gr)).ι.app j)) (Gl.map ((whiskerRight α' Gr).app j))
    (Gl.map (colimit.desc (F' ⋙ Gr) { pt := c.pt, ι := whiskerRight α' Gr ≫ c.ι })) (Gl.map (c.ι.app j))
⊢ c'.ι.app j ≫ (IsColimit.coconePointUniqueUpToIso hl hr).hom =
    β.inv.app j ≫ Gl.map ((colimit.cocone (F' ⋙ Gr)).ι.app j)
case refine_1
J : Type v'
inst✝⁹ : Category.{u', v'} J
C : Type u
inst✝⁸ : Category.{v, u} C
K : Type u_1
inst✝⁷ : Category.{?u.215693, u_1} K
D : Type u_2
inst✝⁶ : Category.{u_3, u_2} D
inst✝⁵ : HasColimitsOfShape J C
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝⁴ : Full Gr
inst✝³ : Faithful Gr
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsVanKampenColimit c
inst✝² : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)
inst✝¹ : (X : D) → (f : X ⟶ Gl.obj c.pt) → PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl
inst✝ : (X : C) → (i : J) → (f : X ⟶ c.pt) → PreservesLimit (cospan f (c.ι.app i)) Gl
this✝² : PreservesLimitsOfSize.{0, 0, u_3, v, u_2, u} Gr
this✝¹ : PreservesColimitsOfSize.{u', v', v, u_3, u, u_2} Gl
F' : J ⥤ D
c' : Cocone F'
α : F' ⟶ (F ⋙ Gr) ⋙ Gl
f : c'.pt ⟶ (Gl.mapCocone c).pt
h : α ≫ (Gl.mapCocone c).ι = c'.ι ≫ (Functor.const J).map f
hα : NatTrans.Equifibered α
hc' : IsColimit c'
j : J
α' : F' ⟶ F := α ≫ (Functor.associator F Gr Gl).hom ≫ whiskerLeft F adj.counit ≫ (Functor.rightUnitor F).hom
hα' : NatTrans.Equifibered α'
hα'' : ∀ (j : J), Gl.map (Gr.map (α'.app j)) = adj.counit.app (F'.toPrefunctor.1 j) ≫ α.app j
β : F' ⋙ Gr ⋙ Gl ≅ F' := isoWhiskerLeft F' (asIso adj.counit) ≪≫ Functor.rightUnitor F'
hl : IsColimit ((Cocones.precompose β.hom).obj c') := (IsColimit.precomposeHomEquiv β c').symm hc'
hr : IsColimit (Gl.mapCocone (colimit.cocone (F' ⋙ Gr))) := isColimitOfPreserves Gl (colimit.isColimit (F' ⋙ Gr))
this✝ : α.app j = β.inv.app j ≫ Gl.map (Gr.map (α'.app j))
this :
  f =
    (IsColimit.coconePointUniqueUpToIso hl hr).hom ≫
      Gl.map (colimit.desc (F' ⋙ Gr) { pt := c.pt, ι := whiskerRight α' Gr ≫ c.ι })
⊢ whiskerRight α' Gr ≫ c.ι =
    (colimit.cocone (F' ⋙ Gr)).ι ≫
      (Functor.const J).map (colimit.desc (F' ⋙ Gr) { pt := c.pt, ι := whiskerRight α' Gr ≫ c.ι })

Next tactic:
rw [← IsIso.inv_comp_eq, ← Category.assoc, NatIso.inv_inv_app]