Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: case refine_1.refine_1
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝¹ E✝ : PreOneHypercover S
F✝ : PreOneHypercover S
G✝ : PreOneHypercover S
E F : PreOneHypercover S
f : E ≅ F
G : Cᵒᵖ ⥤ A
⊢ (i : E.multicospanShape.L) →
    (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.left i) ≅
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.left i)

pp: case refine_1.refine_2
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝¹ E✝ : PreOneHypercover S
F✝ : PreOneHypercover S
G✝ : PreOneHypercover S
E F : PreOneHypercover S
f : E ≅ F
G : Cᵒᵖ ⥤ A
⊢ (i : E.multicospanShape.R) →
    (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.right i) ≅
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.right i)

pp: case refine_1.refine_3
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝¹ E✝ : PreOneHypercover S
F✝ : PreOneHypercover S
G✝ : PreOneHypercover S
E F : PreOneHypercover S
f : E ≅ F
G : Cᵒᵖ ⥤ A
⊢ ∀ (i : E.multicospanShape.R),
    (E.multicospanIndex G).multicospan.map (WalkingMulticospan.Hom.fst i) ≫ Iso.hom (?refine_1.refine_2 i) =
      Iso.hom (?refine_1.refine_1 (E.multicospanShape.fst i)) ≫
        ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).map
          (WalkingMulticospan.Hom.fst i)

pp: case refine_1.refine_4
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝¹ E✝ : PreOneHypercover S
F✝ : PreOneHypercover S
G✝ : PreOneHypercover S
E F : PreOneHypercover S
f : E ≅ F
G : Cᵒᵖ ⥤ A
⊢ ∀ (i : E.multicospanShape.R),
    (E.multicospanIndex G).multicospan.map (WalkingMulticospan.Hom.snd i) ≫ Iso.hom (?refine_1.refine_2 i) =
      Iso.hom (?refine_1.refine_1 (E.multicospanShape.snd i)) ≫
        ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).map
          (WalkingMulticospan.Hom.snd i)

Tactic:
· intro i
      exact G.mapIso (asIso (f.hom.h₀ i)).symm.op

Goal state after:
case refine_1.refine_2
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝¹ E✝ : PreOneHypercover S
F✝ : PreOneHypercover S
G✝ : PreOneHypercover S
E F : PreOneHypercover S
f : E ≅ F
G : Cᵒᵖ ⥤ A
⊢ (i : E.multicospanShape.R) →
    (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.right i) ≅
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.right i)

case refine_1.refine_3
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝¹ E✝ : PreOneHypercover S
F✝ : PreOneHypercover S
G✝ : PreOneHypercover S
E F : PreOneHypercover S
f : E ≅ F
G : Cᵒᵖ ⥤ A
⊢ ∀ (i : E.multicospanShape.R),
    (E.multicospanIndex G).multicospan.map (WalkingMulticospan.Hom.fst i) ≫ Iso.hom (?refine_1.refine_2 i) =
      (G.mapIso (asIso (f.hom.h₀ (E.multicospanShape.fst i))).symm.op).hom ≫
        ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).map
          (WalkingMulticospan.Hom.fst i)

case refine_1.refine_4
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝¹ E✝ : PreOneHypercover S
F✝ : PreOneHypercover S
G✝ : PreOneHypercover S
E F : PreOneHypercover S
f : E ≅ F
G : Cᵒᵖ ⥤ A
⊢ ∀ (i : E.multicospanShape.R),
    (E.multicospanIndex G).multicospan.map (WalkingMulticospan.Hom.snd i) ≫ Iso.hom (?refine_1.refine_2 i) =
      (G.mapIso (asIso (f.hom.h₀ (E.multicospanShape.snd i))).symm.op).hom ≫
        ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).map
          (WalkingMulticospan.Hom.snd i)