-- Mathlib/CategoryTheory/Adhesive.lean
theorem IsPushout.isVanKampen_iff (H : IsPushout f g h i) :
    H.IsVanKampen ↔ IsVanKampenColimit (PushoutCocone.mk h i H.w)

Proof state:
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : H✝.IsVanKampen
F' : WalkingSpan ⥤ C
c' : Cocone F'
α : F' ⟶ span f g
fα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt
eα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα
hα : NatTrans.Equifibered α
this : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right
⊢ (Cocones.precompose (diagramIsoSpan F').inv).obj c' ≅
    PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this

Tactic:
refine Cocones.ext (Iso.refl c'.pt) ?_

Result:
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : H✝.IsVanKampen
F' : WalkingSpan ⥤ C
c' : Cocone F'
α : F' ⟶ span f g
fα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt
eα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα
hα : NatTrans.Equifibered α
this : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right
⊢ ∀ (j : WalkingSpan),
    ((Cocones.precompose (diagramIsoSpan F').inv).obj c').ι.app j ≫ (Iso.refl c'.pt).hom =
      (PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this).ι.app j

Proof state:
case mpr.refine_4
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : IsVanKampenColimit (PushoutCocone.mk h i ⋯)
W' X' Y' Z' : C
f' : W' ⟶ X'
g' : W' ⟶ Y'
h' : X' ⟶ Z'
i' : Y' ⟶ Z'
αW : W' ⟶ W
αX : X' ⟶ X
αY : Y' ⟶ Y
αZ : Z' ⟶ Z
hf : IsPullback f' αW αX f
hg : IsPullback g' αW αY g
hh : CommSq h' αX αZ h
hi : CommSq i' αY αZ i
w : CommSq f' g' h' i'
⊢ NatTrans.Equifibered
    { app := fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY, naturality := ⋯ }

Tactic:
rintro i _ (_ | _ | _)

Result:
case mpr.refine_4.id
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i✝ : Y ⟶ Z
H✝ : IsPushout f g h i✝
H : IsVanKampenColimit (PushoutCocone.mk h i✝ ⋯)
W' X' Y' Z' : C
f' : W' ⟶ X'
g' : W' ⟶ Y'
h' : X' ⟶ Z'
i' : Y' ⟶ Z'
αW : W' ⟶ W
αX : X' ⟶ X
αY : Y' ⟶ Y
αZ : Z' ⟶ Z
hf : IsPullback f' αW αX f
hg : IsPullback g' αW αY g
hh : CommSq h' αX αZ h
hi : CommSq i' αY αZ i✝
w : CommSq f' g' h' i'
i : WalkingSpan
⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.id i))
    ({ app := fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY, naturality := ⋯ }.app i)
    ({ app := fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY, naturality := ⋯ }.app i)
    ((span f g).map (WidePushoutShape.Hom.id i))

case mpr.refine_4.init.left
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : IsVanKampenColimit (PushoutCocone.mk h i ⋯)
W' X' Y' Z' : C
f' : W' ⟶ X'
g' : W' ⟶ Y'
h' : X' ⟶ Z'
i' : Y' ⟶ Z'
αW : W' ⟶ W
αX : X' ⟶ X
αY : Y' ⟶ Y
αZ : Z' ⟶ Z
hf : IsPullback f' αW αX f
hg : IsPullback g' αW αY g
hh : CommSq h' αX αZ h
hi : CommSq i' αY αZ i
w : CommSq f' g' h' i'
⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.init WalkingPair.left))
    ({ app := fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY, naturality := ⋯ }.app none)
    ({ app := fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY, naturality := ⋯ }.app
      (some WalkingPair.left))
    ((span f g).map (WidePushoutShape.Hom.init WalkingPair.left))

case mpr.refine_4.init.right
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : IsVanKampenColimit (PushoutCocone.mk h i ⋯)
W' X' Y' Z' : C
f' : W' ⟶ X'
g' : W' ⟶ Y'
h' : X' ⟶ Z'
i' : Y' ⟶ Z'
αW : W' ⟶ W
αX : X' ⟶ X
αY : Y' ⟶ Y
αZ : Z' ⟶ Z
hf : IsPullback f' αW αX f
hg : IsPullback g' αW αY g
hh : CommSq h' αX αZ h
hi : CommSq i' αY αZ i
w : CommSq f' g' h' i'
⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.init WalkingPair.right))
    ({ app := fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY, naturality := ⋯ }.app none)
    ({ app := fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY, naturality := ⋯ }.app
      (some WalkingPair.right))
    ((span f g).map (WidePushoutShape.Hom.init WalkingPair.right))