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

Goal state before:
pp: case of.of
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁴ : Category.{v_1, u_1} J
inst✝³ : Category.{v_2, u_3} C
inst✝² : Category.{v_3, u_2} K
inst✝¹ : Category.{v_4, u_4} D
F : C ⥤ D
inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J],
    IsConnected J → IsClosedUnderLimitsOfShape (fun f ↦ Equifibered f.hom) J
hJ : ¬IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
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
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (WithTerminal.coneEquiv.functor.obj { pt := G, π := α })
inst : IsConnected (WithTerminal J)
i j : J
⊢ ∀ (f : WithTerminal.of i ⟶ WithTerminal.of j),
    (WithTerminal.liftFromOver.obj c).map f ≫
        (WithTerminal.casesOn (WithTerminal.of j)
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom =
      (WithTerminal.casesOn (WithTerminal.of i)
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom ≫
        (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).map f

pp: case of.star
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁴ : Category.{v_1, u_1} J
inst✝³ : Category.{v_2, u_3} C
inst✝² : Category.{v_3, u_2} K
inst✝¹ : Category.{v_4, u_4} D
F : C ⥤ D
inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J],
    IsConnected J → IsClosedUnderLimitsOfShape (fun f ↦ Equifibered f.hom) J
hJ : ¬IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
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
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (WithTerminal.coneEquiv.functor.obj { pt := G, π := α })
inst : IsConnected (WithTerminal J)
i : J
⊢ ∀ (f : WithTerminal.of i ⟶ WithTerminal.star),
    (WithTerminal.liftFromOver.obj c).map f ≫
        (WithTerminal.casesOn WithTerminal.star
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom =
      (WithTerminal.casesOn (WithTerminal.of i)
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom ≫
        (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).map f

pp: case star.of
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁴ : Category.{v_1, u_1} J
inst✝³ : Category.{v_2, u_3} C
inst✝² : Category.{v_3, u_2} K
inst✝¹ : Category.{v_4, u_4} D
F : C ⥤ D
inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J],
    IsConnected J → IsClosedUnderLimitsOfShape (fun f ↦ Equifibered f.hom) J
hJ : ¬IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
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
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (WithTerminal.coneEquiv.functor.obj { pt := G, π := α })
inst : IsConnected (WithTerminal J)
j : J
⊢ ∀ (f : WithTerminal.star ⟶ WithTerminal.of j),
    (WithTerminal.liftFromOver.obj c).map f ≫
        (WithTerminal.casesOn (WithTerminal.of j)
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom =
      (WithTerminal.casesOn WithTerminal.star
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom ≫
        (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).map f

pp: case star.star
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁴ : Category.{v_1, u_1} J
inst✝³ : Category.{v_2, u_3} C
inst✝² : Category.{v_3, u_2} K
inst✝¹ : Category.{v_4, u_4} D
F : C ⥤ D
inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J],
    IsConnected J → IsClosedUnderLimitsOfShape (fun f ↦ Equifibered f.hom) J
hJ : ¬IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
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
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (WithTerminal.coneEquiv.functor.obj { pt := G, π := α })
inst : IsConnected (WithTerminal J)
⊢ ∀ (f : WithTerminal.star ⟶ WithTerminal.star),
    (WithTerminal.liftFromOver.obj c).map f ≫
        (WithTerminal.casesOn WithTerminal.star
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom =
      (WithTerminal.casesOn WithTerminal.star
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom ≫
        (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).map f

Tactic:
<;>

Goal state after:
case of.of
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁴ : Category.{v_1, u_1} J
inst✝³ : Category.{v_2, u_3} C
inst✝² : Category.{v_3, u_2} K
inst✝¹ : Category.{v_4, u_4} D
F : C ⥤ D
inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J],
    IsConnected J → IsClosedUnderLimitsOfShape (fun f ↦ Equifibered f.hom) J
hJ : ¬IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
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
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (WithTerminal.coneEquiv.functor.obj { pt := G, π := α })
inst : IsConnected (WithTerminal J)
i j : J
⊢ ∀ (f : WithTerminal.of i ⟶ WithTerminal.of j),
    (WithTerminal.liftFromOver.obj c).map f ≫
        (WithTerminal.casesOn (WithTerminal.of j)
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom =
      (WithTerminal.casesOn (WithTerminal.of i)
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom ≫
        (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).map f

case of.star
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁴ : Category.{v_1, u_1} J
inst✝³ : Category.{v_2, u_3} C
inst✝² : Category.{v_3, u_2} K
inst✝¹ : Category.{v_4, u_4} D
F : C ⥤ D
inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J],
    IsConnected J → IsClosedUnderLimitsOfShape (fun f ↦ Equifibered f.hom) J
hJ : ¬IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
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
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (WithTerminal.coneEquiv.functor.obj { pt := G, π := α })
inst : IsConnected (WithTerminal J)
i : J
⊢ ∀ (f : WithTerminal.of i ⟶ WithTerminal.star),
    (WithTerminal.liftFromOver.obj c).map f ≫
        (WithTerminal.casesOn WithTerminal.star
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom =
      (WithTerminal.casesOn (WithTerminal.of i)
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom ≫
        (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).map f

case star.of
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁴ : Category.{v_1, u_1} J
inst✝³ : Category.{v_2, u_3} C
inst✝² : Category.{v_3, u_2} K
inst✝¹ : Category.{v_4, u_4} D
F : C ⥤ D
inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J],
    IsConnected J → IsClosedUnderLimitsOfShape (fun f ↦ Equifibered f.hom) J
hJ : ¬IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
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
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (WithTerminal.coneEquiv.functor.obj { pt := G, π := α })
inst : IsConnected (WithTerminal J)
j : J
⊢ ∀ (f : WithTerminal.star ⟶ WithTerminal.of j),
    (WithTerminal.liftFromOver.obj c).map f ≫
        (WithTerminal.casesOn (WithTerminal.of j)
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom =
      (WithTerminal.casesOn WithTerminal.star
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom ≫
        (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).map f

case star.star
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst✝⁴ : Category.{v_1, u_1} J
inst✝³ : Category.{v_2, u_3} C
inst✝² : Category.{v_3, u_2} K
inst✝¹ : Category.{v_4, u_4} D
F : C ⥤ D
inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J],
    IsConnected J → IsClosedUnderLimitsOfShape (fun f ↦ Equifibered f.hom) J
hJ : ¬IsConnected J
G : Over F
x✝ : limitsOfShape (fun f ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
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
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (WithTerminal.coneEquiv.functor.obj { pt := G, π := α })
inst : IsConnected (WithTerminal J)
⊢ ∀ (f : WithTerminal.star ⟶ WithTerminal.star),
    (WithTerminal.liftFromOver.obj c).map f ≫
        (WithTerminal.casesOn WithTerminal.star
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom =
      (WithTerminal.casesOn WithTerminal.star
            (fun x ↦ Iso.refl ((WithTerminal.liftFromOver.obj c).obj (WithTerminal.of x)))
            (Iso.refl ((WithTerminal.liftFromOver.obj c).obj WithTerminal.star))).hom ≫
        (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).map f