Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits

Goal state before:
pp: case refine_2
J✝ : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : Category.{v_2, u_3} C
inst✝³ : Category.{v_3, u_2} K
inst✝² : Category.{v_4, u_4} D
F : C ⥤ D
inst✝¹ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
J : Type u_1
inst✝ : Category.{v_1, u_1} J
hJ : IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
i j : C
f : i ⟶ j
c : J ⥤ Over F
α : (const J).obj G ⟶ c
hc : IsLimit { pt := G, π := α }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc }.diag.obj j).hom
s : PullbackCone (G.hom.app j) (F.map f)
hcᵢ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj i) hc
hcⱼ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj j).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj j) hc
l : s.pt ⟶ G.left.obj i :=
  hcᵢ.lift
    { pt := s.1, π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }
⊢ l ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).snd = s.snd

pp: case refine_3
J✝ : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : Category.{v_2, u_3} C
inst✝³ : Category.{v_3, u_2} K
inst✝² : Category.{v_4, u_4} D
F : C ⥤ D
inst✝¹ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
J : Type u_1
inst✝ : Category.{v_1, u_1} J
hJ : IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
i j : C
f : i ⟶ j
c : J ⥤ Over F
α : (const J).obj G ⟶ c
hc : IsLimit { pt := G, π := α }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc }.diag.obj j).hom
s : PullbackCone (G.hom.app j) (F.map f)
hcᵢ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj i) hc
hcⱼ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj j).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj j) hc
l : s.pt ⟶ G.left.obj i :=
  hcᵢ.lift
    { pt := s.1, π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }
m : s.pt ⟶ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).pt
hm₁ : m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).fst = s.fst
hm₂ : m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).snd = s.snd
k : J
⊢ (m ≫ ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }).π.app k) ≫
      ({ diag := c, π := α, isLimit := hc }.diag.obj k).left.map f =
    { pt := s.1,
              π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }.π.app
        k ≫
      ({ diag := c, π := α, isLimit := hc }.diag.obj k).left.map f

pp: case refine_4
J✝ : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : Category.{v_2, u_3} C
inst✝³ : Category.{v_3, u_2} K
inst✝² : Category.{v_4, u_4} D
F : C ⥤ D
inst✝¹ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
J : Type u_1
inst✝ : Category.{v_1, u_1} J
hJ : IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
i j : C
f : i ⟶ j
c : J ⥤ Over F
α : (const J).obj G ⟶ c
hc : IsLimit { pt := G, π := α }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc }.diag.obj j).hom
s : PullbackCone (G.hom.app j) (F.map f)
hcᵢ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj i) hc
hcⱼ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj j).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj j) hc
l : s.pt ⟶ G.left.obj i :=
  hcᵢ.lift
    { pt := s.1, π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }
m : s.pt ⟶ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).pt
hm₁ : m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).fst = s.fst
hm₂ : m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).snd = s.snd
k : J
⊢ (m ≫ ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }).π.app k) ≫
      ({ diag := c, π := α, isLimit := hc }.diag.obj k).hom.app i =
    { pt := s.1,
              π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }.π.app
        k ≫
      ({ diag := c, π := α, isLimit := hc }.diag.obj k).hom.app i

Tactic:
· dsimp
    obtain ⟨k⟩ : Nonempty J := inferInstance
    rw [show G.hom.app i = (α.app k).left.app i ≫ (c.obj k).hom.app i by simp [← comp_app]]
    exact (hcᵢ.fac_assoc ..).trans (by simp)

Goal state after:
case refine_3
J✝ : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : Category.{v_2, u_3} C
inst✝³ : Category.{v_3, u_2} K
inst✝² : Category.{v_4, u_4} D
F : C ⥤ D
inst✝¹ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
J : Type u_1
inst✝ : Category.{v_1, u_1} J
hJ : IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
i j : C
f : i ⟶ j
c : J ⥤ Over F
α : (const J).obj G ⟶ c
hc : IsLimit { pt := G, π := α }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc }.diag.obj j).hom
s : PullbackCone (G.hom.app j) (F.map f)
hcᵢ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj i) hc
hcⱼ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj j).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj j) hc
l : s.pt ⟶ G.left.obj i :=
  hcᵢ.lift
    { pt := s.1, π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }
m : s.pt ⟶ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).pt
hm₁ : m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).fst = s.fst
hm₂ : m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).snd = s.snd
k : J
⊢ (m ≫ ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }).π.app k) ≫
      ({ diag := c, π := α, isLimit := hc }.diag.obj k).left.map f =
    { pt := s.1,
              π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }.π.app
        k ≫
      ({ diag := c, π := α, isLimit := hc }.diag.obj k).left.map f

case refine_4
J✝ : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁵ : Category.{v_1, u_1} J✝
inst✝⁴ : Category.{v_2, u_3} C
inst✝³ : Category.{v_3, u_2} K
inst✝² : Category.{v_4, u_4} D
F : C ⥤ D
inst✝¹ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
J : Type u_1
inst✝ : Category.{v_1, u_1} J
hJ : IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
i j : C
f : i ⟶ j
c : J ⥤ Over F
α : (const J).obj G ⟶ c
hc : IsLimit { pt := G, π := α }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc }.diag.obj j).hom
s : PullbackCone (G.hom.app j) (F.map f)
hcᵢ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj i) hc
hcⱼ : IsLimit ((Over.forget F ⋙ (evaluation C D).obj j).mapCone { pt := G, π := α }) :=
  isLimitOfPreserves (Over.forget F ⋙ (evaluation C D).obj j) hc
l : s.pt ⟶ G.left.obj i :=
  hcᵢ.lift
    { pt := s.1, π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }
m : s.pt ⟶ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).pt
hm₁ : m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).fst = s.fst
hm₂ : m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).snd = s.snd
k : J
⊢ (m ≫ ((Over.forget F ⋙ (evaluation C D).obj i).mapCone { pt := G, π := α }).π.app k) ≫
      ({ diag := c, π := α, isLimit := hc }.diag.obj k).hom.app i =
    { pt := s.1,
              π := { app := fun k ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }.π.app
        k ≫
      ({ diag := c, π := α, isLimit := hc }.diag.obj k).hom.app i