-- Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
theorem eq_desc_of_comp_eq (g : widePushout _ _ arrows ⟶ X) :
    (∀ j : J, ι arrows j ≫ g = fs j) → head arrows ≫ g = f → g = desc f fs w

Proof state:
case x.some
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
val✝ : J
⊢ (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).ι.app (some val✝) ≫ g =
    (WidePushoutShape.mkCocone f fs w).ι.app (some val✝)

Tactic:
apply h1

Result:
no goals

Proof state:
case x
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
⊢ ∀ (j : WidePushoutShape J),
    (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).ι.app j ≫ g = (WidePushoutShape.mkCocone f fs w).ι.app j

Tactic:
rintro (_ | _)

Result:
case x.none
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
⊢ (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).ι.app none ≫ g =
    (WidePushoutShape.mkCocone f fs w).ι.app none

case x.some
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
val✝ : J
⊢ (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).ι.app (some val✝) ≫ g =
    (WidePushoutShape.mkCocone f fs w).ι.app (some val✝)

Proof state:
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
⊢ g = desc f fs w

Tactic:
apply
  (colimit.isColimit (WidePushoutShape.wideSpan B objs arrows)).uniq
    (WidePushoutShape.mkCocone f fs <| w)

Result:
case x
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
⊢ ∀ (j : WidePushoutShape J),
    (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).ι.app j ≫ g = (WidePushoutShape.mkCocone f fs w).ι.app j

Proof state:
case x.none
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
⊢ (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).ι.app none ≫ g =
    (WidePushoutShape.mkCocone f fs w).ι.app none

Tactic:
apply h2

Result:
no goals

Proof state:
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
⊢ (∀ (j : J), ι arrows j ≫ g = fs j) → head arrows ≫ g = f → g = desc f fs w

Tactic:
intro h1 h2

Result:
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g = fs j
h2 : head arrows ≫ g = f
⊢ g = desc f fs w