-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
theorem pentagon_bimod {V W X Y Z : Mon_ C} (M : Bimod V W) (N : Bimod W X) (P : Bimod X Y)
    (Q : Bimod Y Z) :
    whiskerRight (associatorBimod M N P).hom Q ≫
      (associatorBimod M (N.tensorBimod P) Q).hom ≫
        whiskerLeft M (associatorBimod N P Q).hom =
      (associatorBimod (M.tensorBimod N) P Q).hom ≫
        (associatorBimod M N (P.tensorBimod Q)).hom

Proof state:
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
      (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
        (M.X ◁ (α_ N.X P.X Q.X).hom ≫
            M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
              M.X ◁
                coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
                  ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
                    N.X ◁ TensorBimod.actLeft P Q)) ≫
          colimit.ι
            (parallelPair
              (M.actRight ▷
                coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                  ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
              ((α_ M.X W.X
                    (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                      ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
                M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
            WalkingParallelPair.one =
    (α_ (M.X ⊗ N.X) P.X Q.X).hom ≫
      (((M.X ⊗ N.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
            coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷
              coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
          (PreservesCoequalizer.iso
              (tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))) (M.actRight ▷ N.X)
              ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv) ≫
        coequalizer.desc
          ((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
            M.X ◁
                coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
                  ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
                    N.X ◁ TensorBimod.actLeft P Q) ≫
              coequalizer.π
                (M.actRight ▷
                  coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                    ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
                ((α_ M.X W.X
                      (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                        ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
                  M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
          ⋯

Tactic:
slice_rhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]

Result:
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
      (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
        (M.X ◁ (α_ N.X P.X Q.X).hom ≫
            M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
              M.X ◁
                coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
                  ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
                    N.X ◁ TensorBimod.actLeft P Q)) ≫
          colimit.ι
            (parallelPair
              (M.actRight ▷
                coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                  ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
              ((α_ M.X W.X
                    (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                      ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
                M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
            WalkingParallelPair.one =
    (α_ (M.X ⊗ N.X) P.X Q.X).hom ≫
      (M.X ⊗ N.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
        (α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
          M.X ◁
              coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
                ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
                  N.X ◁ TensorBimod.actLeft P Q) ≫
            coequalizer.π
              (M.actRight ▷
                coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                  ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
              ((α_ M.X W.X
                    (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                      ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
                M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))