theorem IsUniversalColimit.map_reflective
    {Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Full Gr] [Faithful Gr]
    {F : J ⥤ D} {c : Cocone (F ⋙ Gr)}
    (H : IsUniversalColimit 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] :
    IsUniversalColimit (Gl.mapCocone c) := by
  have := adj.rightAdjointPreservesLimits
  have : PreservesColimitsOfSize.{u', v'} Gl := adj.leftAdjointPreservesColimits
  intros F' c' α f h hα hc'
  have : HasPullback (Gl.map (Gr.map f)) (Gl.map (adj.unit.app c.pt)) :=
    ⟨⟨_, isLimitPullbackConeMapOfIsLimit _ pullback.condition
      (IsPullback.of_hasPullback _ _).isLimit⟩⟩
  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 hadj : ∀ X, Gl.map (adj.unit.app X) = inv (adj.counit.app _)
  · intro X
    apply IsIso.eq_inv_of_inv_hom_id
    exact adj.left_triangle_components
  haveI : ∀ X, IsIso (Gl.map (adj.unit.app X)) := by
    simp_rw [hadj]
    infer_instance
  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]
  have hc'' : ∀ j, α.app j ≫ Gl.map (c.ι.app j) = c'.ι.app j ≫ f := NatTrans.congr_app h
  let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
  let c'' : Cocone (F' ⋙ Gr)
  · refine
    { pt := pullback (Gr.map f) (adj.unit.app _)
      ι := { app := λ j ↦ pullback.lift (Gr.map $ c'.ι.app j) (Gr.map (α'.app j) ≫ c.ι.app j) ?_
             naturality := ?_ } }
    · 

Proof state:
case c''.refine_1
J : Type v'
inst✝⁷ : Category.{u', v'} J
C : Type u
inst✝⁶ : Category.{v, u} C
K : Type u_1
inst✝⁵ : Category.{?u.159962, u_1} K
D : Type u_2
inst✝⁴ : Category.{u_3, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Full Gr
inst✝² : Faithful Gr
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsUniversalColimit c
inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)
inst✝ : (X : D) → (f : X ⟶ Gl.obj c.pt) → PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl
this✝² : PreservesLimitsOfSize.{?u.163214, ?u.163213, 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' : ∀ (j : J), IsPullback (c'.ι.app j) (α.app j) f ((Gl.mapCocone c).ι.app j)
this✝ : HasPullback (Gl.map (Gr.map f)) (Gl.map (adj.unit.app c.pt))
α' : F' ⟶ F := α ≫ (Functor.associator F Gr Gl).hom ≫ whiskerLeft F adj.counit ≫ (Functor.rightUnitor F).hom
hα' : NatTrans.Equifibered α'
hadj : ∀ (X : C), Gl.map (adj.unit.app X) = inv (adj.counit.app (Gl.toPrefunctor.1 ((𝟭 C).obj X)))
this : ∀ (X : C), IsIso (Gl.map (adj.unit.app X))
hα'' : ∀ (j : J), Gl.map (Gr.map (α'.app j)) = adj.counit.app (F'.toPrefunctor.1 j) ≫ α.app j
hc'' : ∀ (j : J), α.app j ≫ Gl.map (c.ι.app j) = c'.ι.app j ≫ f
β : F' ⋙ Gr ⋙ Gl ≅ F' := isoWhiskerLeft F' (asIso adj.counit) ≪≫ Functor.rightUnitor F'
j : J
⊢ Gr.map (c'.ι.app j) ≫ Gr.map f = (Gr.map (α'.app j) ≫ c.ι.app j) ≫ adj.unit.app c.pt

Next tactic:
rw [← Gr.map_comp, ← hc'']