Module: Mathlib.CategoryTheory.Monoidal.PushoutProduct

Goal state before:
pp: case refine_1.refine_4.h₀
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasPushouts C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ : Arrow C
W : C
inst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)
inst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.right)
inst✝¹ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.left)
inst✝ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.right)
⊢ (tensorLeft X₁.right).map
        (pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom)) ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
          (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
        pushout.desc
            (Arrow.Hom.left (whiskerLeftIso X₂ X₃).hom ≫
              pushout.desc
                (pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
                (pushout.inl (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                    X₃.right ≫
                  pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                    (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
                ⋯)
            ((α_ X₁.left X₂.right X₃.right).inv ≫
              pushout.inr (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                  X₃.right ≫
                pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
            ⋯ ≫
          pushout.desc
            ((α_ X₁.right X₂.right X₃.left).hom ≫
              X₁.right ◁
                  pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left)
                    (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
            (Arrow.Hom.left (whiskerRightIso X₁ X₂).hom ≫
              pushout.desc
                (X₁.right ◁
                    pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left)
                      (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                  pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                    (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
                (pushout.inr (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
                ⋯)
            ⋯ =
    (tensorLeft X₁.right).map
        (pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom)) ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
          (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
        𝟙 (X₁ □ X₂ □ X₃).left

pp: case refine_1.refine_4.h₁
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasPushouts C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ : Arrow C
W : C
inst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)
inst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.right)
inst✝¹ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.left)
inst✝ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.right)
⊢ (tensorLeft X₁.right).map
        (pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom)) ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
          (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
        pushout.desc
            (Arrow.Hom.left (whiskerLeftIso X₂ X₃).hom ≫
              pushout.desc
                (pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
                (pushout.inl (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                    X₃.right ≫
                  pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                    (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
                ⋯)
            ((α_ X₁.left X₂.right X₃.right).inv ≫
              pushout.inr (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                  X₃.right ≫
                pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
            ⋯ ≫
          pushout.desc
            ((α_ X₁.right X₂.right X₃.left).hom ≫
              X₁.right ◁
                  pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left)
                    (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
            (Arrow.Hom.left (whiskerRightIso X₁ X₂).hom ≫
              pushout.desc
                (X₁.right ◁
                    pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left)
                      (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                  pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                    (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
                (pushout.inr (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
                ⋯)
            ⋯ =
    (tensorLeft X₁.right).map
        (pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom)) ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
          (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
        𝟙 (X₁ □ X₂ □ X₃).left

Tactic:
<;>

Goal state after:
case refine_1.refine_4.h₀
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasPushouts C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ : Arrow C
W : C
inst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)
inst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.right)
inst✝¹ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.left)
inst✝ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.right)
⊢ (tensorLeft X₁.right).map
        (pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom)) ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
          (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
        pushout.desc
            (Arrow.Hom.left (whiskerLeftIso X₂ X₃).hom ≫
              pushout.desc
                (pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
                (pushout.inl (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                    X₃.right ≫
                  pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                    (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
                ⋯)
            ((α_ X₁.left X₂.right X₃.right).inv ≫
              pushout.inr (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                  X₃.right ≫
                pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
            ⋯ ≫
          pushout.desc
            ((α_ X₁.right X₂.right X₃.left).hom ≫
              X₁.right ◁
                  pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left)
                    (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
            (Arrow.Hom.left (whiskerRightIso X₁ X₂).hom ≫
              pushout.desc
                (X₁.right ◁
                    pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left)
                      (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                  pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                    (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
                (pushout.inr (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
                ⋯)
            ⋯ =
    (tensorLeft X₁.right).map
        (pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom)) ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
          (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
        𝟙 (X₁ □ X₂ □ X₃).left

case refine_1.refine_4.h₁
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasPushouts C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ : Arrow C
W : C
inst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)
inst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.right)
inst✝¹ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.left)
inst✝ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.right)
⊢ (tensorLeft X₁.right).map
        (pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom)) ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
          (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
        pushout.desc
            (Arrow.Hom.left (whiskerLeftIso X₂ X₃).hom ≫
              pushout.desc
                (pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
                (pushout.inl (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                    X₃.right ≫
                  pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                    (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
                ⋯)
            ((α_ X₁.left X₂.right X₃.right).inv ≫
              pushout.inr (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                  X₃.right ≫
                pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
            ⋯ ≫
          pushout.desc
            ((α_ X₁.right X₂.right X₃.left).hom ≫
              X₁.right ◁
                  pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left)
                    (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
            (Arrow.Hom.left (whiskerRightIso X₁ X₂).hom ≫
              pushout.desc
                (X₁.right ◁
                    pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left)
                      (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                  pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                    (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
                (pushout.inr (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
                ⋯)
            ⋯ =
    (tensorLeft X₁.right).map
        (pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom)) ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
          (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
        𝟙 (X₁ □ X₂ □ X₃).left