-- Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : Type u₂
inst_1 : Category.{v₂, u₂} J
X Y Z : C
f_1 : X ⟶ Y
g : X ⟶ Z
f : Option.some WalkingPair.right ⟶ none
some :
  Eq (α :=
    obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) (Option.some WalkingPair.right) ⟶
      (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj none)
    (map (cospan f_1.op g.op) (WidePullbackShape.Hom.term WalkingPair.right) ≫
      Iso.hom
        ((fun (x : WalkingCospan) ↦
            (match (motive :=
                (x : WalkingCospan) →
                  obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) x ≅ (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj x)
                x with
              | none => Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) none)
              | Option.some WalkingPair.left => Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) WalkingCospan.left)
              | Option.some WalkingPair.right =>
                Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) WalkingCospan.right) :
              obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) x ≅ (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj x))
          none))
    (Iso.hom
        ((fun (x : WalkingCospan) ↦
            (match (motive :=
                (x : WalkingCospan) →
                  obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) x ≅ (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj x)
                x with
              | none => Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) none)
              | Option.some WalkingPair.left => Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) WalkingCospan.left)
              | Option.some WalkingPair.right =>
                Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) WalkingCospan.right) :
              obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) x ≅ (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj x))
          (Option.some WalkingPair.right)) ≫
      (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).map (WidePullbackShape.Hom.term WalkingPair.right))
⊢ Eq (α :=
    obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) (Option.some WalkingPair.right) ⟶
      (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj none)
    (map (cospan f_1.op g.op) f ≫
      Iso.hom
        ((fun (x : WalkingCospan) ↦
            (match (motive :=
                (x : WalkingCospan) →
                  obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) x ≅ (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj x)
                x with
              | none => Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) none)
              | Option.some WalkingPair.left => Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) WalkingCospan.left)
              | Option.some WalkingPair.right =>
                Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) WalkingCospan.right) :
              obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) x ≅ (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj x))
          none))
    (Iso.hom
        ((fun (x : WalkingCospan) ↦
            (match (motive :=
                (x : WalkingCospan) →
                  obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) x ≅ (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj x)
                x with
              | none => Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) none)
              | Option.some WalkingPair.left => Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) WalkingCospan.left)
              | Option.some WalkingPair.right =>
                Iso.refl (obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) WalkingCospan.right) :
              obj.{0, v₁, 0, u₁} (cospan f_1.op g.op) x ≅ (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).obj x))
          (Option.some WalkingPair.right)) ≫
      (walkingSpanOpEquiv.inverse ⋙ (span f_1 g).op).map f)

Tactic:
cases f