Module: Mathlib.CategoryTheory.Adhesive.Basic

Goal state before:
pp: case i'_w.i'_w
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
W X Y Z✝ : C
f✝ : W ⟶ X
g✝ : W ⟶ Y
h : X ⟶ Z✝
i : Y ⟶ Z✝
inst✝² : Adhesive C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst✝¹ : Mono a
inst✝ : Mono b
K : C
f g : K ⟶ pushout (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (fst a b) (snd a b) := inr (fst a b) (snd a b)
this✝¹ : Mono u := mono_of_isPushout_of_mono_right (of_hasPushout (fst a b) (snd a b))
this✝ : Mono v := mono_of_isPushout_of_mono_left (of_hasPushout (fst a b) (snd a b))
sq_f_u : IsPullback (fst f u) (snd f u) f u := of_hasPullback f u
sq_f_v : IsPullback (fst f v) (snd f v) f v := of_hasPullback f v
sq_g_u : IsPullback (fst g u) (snd g u) g u := of_hasPullback g u
sq_g_v : IsPullback (fst g v) (snd g v) g v := of_hasPullback g v
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w✝² : C
f' : w✝² ⟶ pullback f u
w✝¹ : w✝² ⟶ pullback f v
w✝ : w✝² ⟶ pullback a b
p₁ : IsPullback f' w✝ (snd f u) (fst a b)
left✝ : IsPullback w✝¹ w✝ (snd f v) (snd a b)
h₁ : IsPushout f' w✝¹ (fst f u) (fst f v)
this : Mono f' := Eq.mpr (id (congrArg (fun _a ↦ Mono _a) (Eq.symm (isoPullback_hom_fst p₁)))) inferInstance
sq₂₁ : IsPullback (fst m₂ l₁) (snd m₂ l₁) m₂ l₁ := of_hasPullback m₂ l₁
sq₂₂ : IsPullback (fst m₂ l₂) (snd m₂ l₂) m₂ l₂ := of_hasPullback m₂ l₂
⊢ (snd m₂ l₂ ≫ snd f v) ≫ v = (fst m₂ l₂ ≫ snd g v) ≫ v

Tactic:
refine ?_ =≫ v

Goal state after:
case i'_w.i'_w
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
W X Y Z✝ : C
f✝ : W ⟶ X
g✝ : W ⟶ Y
h : X ⟶ Z✝
i : Y ⟶ Z✝
inst✝² : Adhesive C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst✝¹ : Mono a
inst✝ : Mono b
K : C
f g : K ⟶ pushout (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (fst a b) (snd a b) := inr (fst a b) (snd a b)
this✝¹ : Mono u := mono_of_isPushout_of_mono_right (of_hasPushout (fst a b) (snd a b))
this✝ : Mono v := mono_of_isPushout_of_mono_left (of_hasPushout (fst a b) (snd a b))
sq_f_u : IsPullback (fst f u) (snd f u) f u := of_hasPullback f u
sq_f_v : IsPullback (fst f v) (snd f v) f v := of_hasPullback f v
sq_g_u : IsPullback (fst g u) (snd g u) g u := of_hasPullback g u
sq_g_v : IsPullback (fst g v) (snd g v) g v := of_hasPullback g v
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w✝² : C
f' : w✝² ⟶ pullback f u
w✝¹ : w✝² ⟶ pullback f v
w✝ : w✝² ⟶ pullback a b
p₁ : IsPullback f' w✝ (snd f u) (fst a b)
left✝ : IsPullback w✝¹ w✝ (snd f v) (snd a b)
h₁ : IsPushout f' w✝¹ (fst f u) (fst f v)
this : Mono f' := Eq.mpr (id (congrArg (fun _a ↦ Mono _a) (Eq.symm (isoPullback_hom_fst p₁)))) inferInstance
sq₂₁ : IsPullback (fst m₂ l₁) (snd m₂ l₁) m₂ l₁ := of_hasPullback m₂ l₁
sq₂₂ : IsPullback (fst m₂ l₂) (snd m₂ l₂) m₂ l₂ := of_hasPullback m₂ l₂
⊢ snd m₂ l₂ ≫ snd f v = fst m₂ l₂ ≫ snd g v