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 := ?_ } }
    · rw [← Gr.map_comp, ← hc'']
      erw [← adj.unit_naturality]
      rw [Gl.map_comp, hα'']
      dsimp
      simp only [Category.assoc, Functor.map_comp, adj.right_triangle_components_assoc]
    · intros i j g
      dsimp
      ext
      all_goals simp only [Category.comp_id, Category.id_comp, Category.assoc,
        ← Functor.map_comp, pullback.lift_fst, pullback.lift_snd, ← Functor.map_comp_assoc]
      · congr 1
        exact c'.w _
      · rw [α.naturality_assoc]
        dsimp
        rw [adj.counit_naturality, ← Category.assoc, Gr.map_comp_assoc]
        congr 1
        exact c.w _
  let cf : (Cocones.precompose β.hom).obj c' ⟶ Gl.mapCocone c''
  · refine { hom := pullback.lift ?_ f ?_ ≫ (PreservesPullback.iso _ _ _).inv, w := ?_ }
    exact (inv $ adj.counit.app c'.pt)
    · rw [IsIso.inv_comp_eq, ← adj.counit_naturality_assoc f, ← cancel_mono (adj.counit.app $
        Gl.obj c.pt), Category.assoc, Category.assoc, adj.left_triangle_components]
      erw [Category.comp_id]
      rfl
    · intro j
      rw [← Category.assoc, Iso.comp_inv_eq]
      ext
      all_goals simp only [PreservesPullback.iso_hom_fst, PreservesPullback.iso_hom_snd,
          pullback.lift_fst, pullback.lift_snd, Category.assoc,
          Functor.mapCocone_ι_app, ← Gl.map_comp]
      · rw [IsIso.comp_inv_eq, adj.counit_naturality]
        dsimp
        rw [Category.comp_id]
      · rw [Gl.map_comp, hα'', Category.assoc, hc'']
        dsimp
        rw [Category.comp_id, Category.assoc]
  have : cf.hom ≫ (PreservesPullback.iso _ _ _).hom ≫ pullback.fst ≫ adj.counit.app _ = 𝟙 _
  · simp only [IsIso.inv_hom_id, Iso.inv_hom_id_assoc, Category.assoc, pullback.lift_fst_assoc]
  have : IsIso cf
  · apply @Cocones.cocone_iso_of_hom_iso (i := ?_)
    rw [← IsIso.eq_comp_inv] at this
    rw [this]
    infer_instance
  have ⟨Hc''⟩ := H c'' (whiskerRight α' Gr) pullback.snd ?_ (hα'.whiskerRight Gr) ?_
  · exact ⟨IsColimit.precomposeHomEquiv β c' $
      (isColimitOfPreserves Gl Hc'').ofIsoColimit (asIso cf).symm⟩
  · ext j
    dsimp
    simp only [Category.comp_id, Category.id_comp, Category.assoc,
      Functor.map_comp, pullback.lift_snd]
  · intro j
    apply IsPullback.of_right _ _ (IsPullback.of_hasPullback _ _)
    · dsimp
      simp only [Category.comp_id, Category.id_comp, Category.assoc, Functor.map_comp,
        pullback.lift_fst]
      rw [← Category.comp_id (Gr.map f)]
      

Proof state:
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✝⁴ : PreservesLimitsOfSize.{0, 0, u_3, v, u_2, u} Gr
this✝³ : 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✝² : 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✝¹ : ∀ (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'
c'' : Cocone (F' ⋙ Gr) :=
  { pt := pullback (Gr.map f) (adj.unit.app c.pt),
    ι :=
      NatTrans.mk fun j =>
        pullback.lift (Gr.map (c'.ι.app j)) (Gr.map (α'.app j) ≫ c.ι.app j)
          (_ : Gr.map (c'.ι.app j) ≫ Gr.map f = (Gr.map (α'.app j) ≫ c.ι.app j) ≫ adj.unit.app c.pt) }
cf : (Cocones.precompose β.hom).obj c' ⟶ Gl.mapCocone c'' :=
  CoconeMorphism.mk
    (pullback.lift (inv (adj.counit.app c'.pt)) f
        (_ : inv (adj.counit.app c'.pt) ≫ Gl.map (Gr.map f) = f ≫ Gl.map (adj.unit.app c.pt)) ≫
      (PreservesPullback.iso Gl (Gr.map f) (adj.unit.app c.pt)).inv)
this✝ :
  cf.hom ≫ (PreservesPullback.iso Gl (Gr.map f) (adj.unit.app c.pt)).hom ≫ pullback.fst ≫ adj.counit.app c'.pt =
    𝟙 ((Cocones.precompose β.hom).obj c').pt
this : IsIso cf
j : J
⊢ IsPullback (Gr.map (c'.ι.app j)) (Gr.map (α.app j) ≫ Gr.map (adj.counit.app (F.obj j)))
    (Gr.map f ≫ 𝟙 (Gr.obj (Gl.mapCocone c).pt)) (c.ι.app j ≫ adj.unit.app c.pt)

Next tactic:
refine ((hc' j).map Gr).paste_vert (IsPullback.of_vert_isIso ⟨?_⟩)