Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: J : Type v'
inst✝⁸ : Category.{u', v'} J
C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u_2
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : HasColimitsOfShape J C
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝⁴ : Gr.Full
inst✝³ : Gr.Faithful
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✝¹ : PreservesLimitsOfSize.{?u.178, ?u.177, v_2, v, u_2, u} Gr
this✝ : PreservesColimitsOfSize.{u', v', v, v_2, 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 ≫ (const J).map f
hα : NatTrans.Equifibered α
hc' : IsColimit c'
j : J
α' : F' ⟶ F := α ≫ (F.associator Gr Gl).hom ≫ F.whiskerLeft adj.counit ≫ F.rightUnitor.hom
hα' : NatTrans.Equifibered α'
hα'' : ∀ (j : J), Gl.map (Gr.map (α'.app j)) = adj.counit.app (F'.1 j) ≫ α.app j
β : F' ⋙ Gr ⋙ Gl ≅ F' := F'.isoWhiskerLeft (asIso adj.counit) ≪≫ F'.rightUnitor
hl : IsColimit ((Cocone.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))
⊢ IsPullback (c'.ι.app j) (β.inv.app j ≫ Gl.map (Gr.map (α'.app j))) f ((Gl.mapCocone c).ι.app j)

Tactic:
have : f = (hl.coconePointUniqueUpToIso hr).hom ≫
    Gl.map (colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩) := by
    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 [β]
      simp only [Category.comp_id, hα'', Category.assoc, Gl.map_comp]
      congr 1
      exact (NatTrans.congr_app h j).symm

Goal state after:
J : Type v'
inst✝⁸ : Category.{u', v'} J
C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u_2
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : HasColimitsOfShape J C
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝⁴ : Gr.Full
inst✝³ : Gr.Faithful
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.{?u.178, ?u.177, v_2, v, u_2, u} Gr
this✝¹ : PreservesColimitsOfSize.{u', v', v, v_2, 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 ≫ (const J).map f
hα : NatTrans.Equifibered α
hc' : IsColimit c'
j : J
α' : F' ⟶ F := α ≫ (F.associator Gr Gl).hom ≫ F.whiskerLeft adj.counit ≫ F.rightUnitor.hom
hα' : NatTrans.Equifibered α'
hα'' : ∀ (j : J), Gl.map (Gr.map (α'.app j)) = adj.counit.app (F'.1 j) ≫ α.app j
β : F' ⋙ Gr ⋙ Gl ≅ F' := F'.isoWhiskerLeft (asIso adj.counit) ≪≫ F'.rightUnitor
hl : IsColimit ((Cocone.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 =
    (hl.coconePointUniqueUpToIso hr).hom ≫ Gl.map (colimit.desc (F' ⋙ Gr) { pt := c.pt, ι := whiskerRight α' Gr ≫ c.ι })
⊢ IsPullback (c'.ι.app j) (β.inv.app j ≫ Gl.map (Gr.map (α'.app j))) f ((Gl.mapCocone c).ι.app j)