Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: J : Type v'
inst✝⁶ : Category.{u', v'} J
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_2
inst✝⁴ : Category.{v_2, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
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.146, ?u.145, 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' : ∀ (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 ≫ (F.associator Gr Gl).hom ≫ F.whiskerLeft adj.counit ≫ F.rightUnitor.hom
hα' : NatTrans.Equifibered α'
hadj : ∀ (X : C), Gl.map (adj.unit.app X) = inv (adj.counit.app (Gl.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'.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' := F'.isoWhiskerLeft (asIso adj.counit) ≪≫ F'.rightUnitor
c'' : Cocone (F' ⋙ Gr) :=
  { pt := pullback (Gr.map f) (adj.unit.app c.pt),
    ι := { app := fun j ↦ pullback.lift (Gr.map (c'.ι.app j)) (Gr.map (α'.app j) ≫ c.ι.app j) ⋯, naturality := ⋯ } }
⊢ Nonempty (IsColimit c')

Tactic:
let cf : (Cocone.precompose β.hom).obj c' ⟶ Gl.mapCocone c'' := by
    refine { hom := pullback.lift ?_ f ?_ ≫ (PreservesPullback.iso _ _ _).inv, w := ?_ }
    · exact inv <| adj.counit.app c'.pt
    · simp [← cancel_mono (adj.counit.app <| Gl.obj c.pt)]
    · intro j
      rw [← Category.assoc, Iso.comp_inv_eq]
      ext
      all_goals simp only [c'', PreservesPullback.iso_hom_fst, PreservesPullback.iso_hom_snd,
          pullback.lift_fst, pullback.lift_snd, Category.assoc,
          Functor.mapCocone_ι_app, ← Gl.map_comp]
      · dsimp [β]
        simp only [IsIso.comp_inv_eq, adj.counit_naturality, Category.comp_id]
      · rw [Gl.map_comp, hα'', Category.assoc, hc'']
        dsimp [β]
        rw [Category.comp_id, Category.assoc]

Goal state after:
J : Type v'
inst✝⁶ : Category.{u', v'} J
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_2
inst✝⁴ : Category.{v_2, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
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.146, ?u.145, 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' : ∀ (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 ≫ (F.associator Gr Gl).hom ≫ F.whiskerLeft adj.counit ≫ F.rightUnitor.hom
hα' : NatTrans.Equifibered α'
hadj : ∀ (X : C), Gl.map (adj.unit.app X) = inv (adj.counit.app (Gl.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'.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' := F'.isoWhiskerLeft (asIso adj.counit) ≪≫ F'.rightUnitor
c'' : Cocone (F' ⋙ Gr) :=
  { pt := pullback (Gr.map f) (adj.unit.app c.pt),
    ι := { app := fun j ↦ pullback.lift (Gr.map (c'.ι.app j)) (Gr.map (α'.app j) ≫ c.ι.app j) ⋯, naturality := ⋯ } }
cf : (Cocone.precompose β.hom).obj c' ⟶ Gl.mapCocone c'' :=
  {
    hom :=
      pullback.lift (inv (adj.counit.app c'.pt)) f ⋯ ≫ (PreservesPullback.iso Gl (Gr.map f) (adj.unit.app c.pt)).inv,
    w := ⋯ }
⊢ Nonempty (IsColimit c')