-- Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
theorem hom_ext (g1 g2 : X ⟶ widePullback _ _ arrows) : (∀ j : J,
    g1 ≫ π arrows j = g2 ≫ π arrows j) → g1 ≫ base arrows = g2 ≫ base arrows → g1 = g2

Proof state:
case w
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
h1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j
h2 : g1 ≫ base arrows = g2 ≫ base arrows
⊢ ∀ (j : WidePullbackShape J),
    g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) j =
      g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) j

Tactic:
rintro (_ | _)

Result:
case w.none
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
h1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j
h2 : g1 ≫ base arrows = g2 ≫ base arrows
⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) none =
    g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) none

case w.some
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
h1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j
h2 : g1 ≫ base arrows = g2 ≫ base arrows
val✝ : J
⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) (some val✝) =
    g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) (some val✝)

Proof state:
case w.some
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
h1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j
h2 : g1 ≫ base arrows = g2 ≫ base arrows
val✝ : J
⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) (some val✝) =
    g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) (some val✝)

Tactic:
apply h1

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) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
⊢ (∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j) → g1 ≫ base arrows = g2 ≫ base arrows → g1 = g2

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) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
h1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j
h2 : g1 ≫ base arrows = g2 ≫ base arrows
⊢ g1 = g2

Proof state:
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
h1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j
h2 : g1 ≫ base arrows = g2 ≫ base arrows
⊢ g1 = g2

Tactic:
apply limit.hom_ext

Result:
case w
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
h1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j
h2 : g1 ≫ base arrows = g2 ≫ base arrows
⊢ ∀ (j : WidePullbackShape J),
    g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) j =
      g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) j

Proof state:
case w.none
J : Type w
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → objs j ⟶ B
inst✝ : HasWidePullback B objs arrows
X : C
g1 g2 : X ⟶ widePullback B objs arrows
h1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j
h2 : g1 ≫ base arrows = g2 ≫ base arrows
⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) none =
    g2 ≫ limit.π (WidePullbackShape.wideCospan B objs arrows) none

Tactic:
apply h2

Result:
no goals