theorem triangle_bimod {X Y Z : Mon_ C} (M : Bimod X Y) (N : Bimod Y Z) :
    (associatorBimod M (regular Y) N).hom ≫ tensorHom (𝟙 M) (leftUnitorBimod N).hom =
      tensorHom (rightUnitorBimod M).hom (𝟙 N) := by
  dsimp [tensorHom, associatorBimod, leftUnitorBimod, rightUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  dsimp [AssociatorBimod.hom]
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.homAux]
  slice_rhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [RightUnitorBimod.hom]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp [regular]
  slice_lhs 1 3 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  

Proof state:
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
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)
X Y Z : Mon_ C
M : Bimod X Y
N : Bimod Y Z
⊢ ((α_ M.X Y.X N.X).hom ≫
        (𝟙 M.X ⊗ coequalizer.π (Y.mul ⊗ 𝟙 N.X) ((α_ Y.X Y.X N.X).hom ≫ (𝟙 Y.X ⊗ N.actLeft))) ≫
          coequalizer.π (M.actRight ⊗ 𝟙 (TensorBimod.X (mk Y.X Y.mul Y.mul) N))
            ((α_ M.X Y.X (TensorBimod.X (mk Y.X Y.mul Y.mul) N)).hom ≫
              (𝟙 M.X ⊗ TensorBimod.actLeft (mk Y.X Y.mul Y.mul) N))) ≫
      colimMap
        (parallelPairHom (M.actRight ⊗ 𝟙 (TensorBimod.X (mk Y.X Y.mul Y.mul) N))
          ((α_ M.X Y.X (TensorBimod.X (mk Y.X Y.mul Y.mul) N)).hom ≫
            (𝟙 M.X ⊗ TensorBimod.actLeft (mk Y.X Y.mul Y.mul) N))
          (M.actRight ⊗ 𝟙 N.X) ((α_ M.X Y.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ((𝟙 M.X ⊗ 𝟙 Y.X) ⊗ LeftUnitorBimod.hom N)
          (𝟙 M.X ⊗ LeftUnitorBimod.hom N)
          (_ :
            (M.actRight ⊗ 𝟙 (tensorBimod (mk Y.X Y.mul Y.mul) N).X) ≫
                ((𝟙 M).hom ⊗
                  (isoOfIso (Iso.mk (LeftUnitorBimod.hom N) (LeftUnitorBimod.inv N))
                        (_ :
                          (tensorBimod (regular Y) N).actLeft ≫ LeftUnitorBimod.hom N =
                            (𝟙 Y.X ⊗ LeftUnitorBimod.hom N) ≫ N.actLeft)
                        (_ :
                          (tensorBimod (regular Y) N).actRight ≫ LeftUnitorBimod.hom N =
                            (LeftUnitorBimod.hom N ⊗ 𝟙 Z.X) ≫ N.actRight)).hom.hom) =
              (((𝟙 M).hom ⊗ 𝟙 Y.X) ⊗
                  (isoOfIso (Iso.mk (LeftUnitorBimod.hom N) (LeftUnitorBimod.inv N))
                        (_ :
                          (tensorBimod (regular Y) N).actLeft ≫ LeftUnitorBimod.hom N =
                            (𝟙 Y.X ⊗ LeftUnitorBimod.hom N) ≫ N.actLeft)
                        (_ :
                          (tensorBimod (regular Y) N).actRight ≫ LeftUnitorBimod.hom N =
                            (LeftUnitorBimod.hom N ⊗ 𝟙 Z.X) ≫ N.actRight)).hom.hom) ≫
                (M.actRight ⊗ 𝟙 N.X))
          (_ :
            ((α_ M.X Y.X (tensorBimod (mk Y.X Y.mul Y.mul) N).X).hom ≫
                  (𝟙 M.X ⊗ (tensorBimod (mk Y.X Y.mul Y.mul) N).actLeft)) ≫
                ((𝟙 M).hom ⊗
                  (isoOfIso (Iso.mk (LeftUnitorBimod.hom N) (LeftUnitorBimod.inv N))
                        (_ :
                          (tensorBimod (regular Y) N).actLeft ≫ LeftUnitorBimod.hom N =
                            (𝟙 Y.X ⊗ LeftUnitorBimod.hom N) ≫ N.actLeft)
                        (_ :
                          (tensorBimod (regular Y) N).actRight ≫ LeftUnitorBimod.hom N =
                            (LeftUnitorBimod.hom N ⊗ 𝟙 Z.X) ≫ N.actRight)).hom.hom) =
              (((𝟙 M).hom ⊗ 𝟙 Y.X) ⊗
                  (isoOfIso (Iso.mk (LeftUnitorBimod.hom N) (LeftUnitorBimod.inv N))
                        (_ :
                          (tensorBimod (regular Y) N).actLeft ≫ LeftUnitorBimod.hom N =
                            (𝟙 Y.X ⊗ LeftUnitorBimod.hom N) ≫ N.actLeft)
                        (_ :
                          (tensorBimod (regular Y) N).actRight ≫ LeftUnitorBimod.hom N =
                            (LeftUnitorBimod.hom N ⊗ 𝟙 Z.X) ≫ N.actRight)).hom.hom) ≫
                (α_ M.X Y.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft))) =
    (coequalizer.π (M.actRight ⊗ 𝟙 Y.X) ((α_ M.X Y.X Y.X).hom ≫ (𝟙 M.X ⊗ Y.mul)) ⊗ 𝟙 N.X) ≫
      (coequalizer.desc M.actRight
            (_ :
              (M.actRight ⊗ 𝟙 (regular Y).X) ≫ M.actRight =
                ((α_ M.X Y.X (regular Y).X).hom ≫ (𝟙 M.X ⊗ (regular Y).actLeft)) ≫ M.actRight) ⊗
          𝟙 N.X) ≫
        colimit.ι (parallelPair (M.actRight ⊗ 𝟙 N.X) ((α_ M.X Y.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)))
          WalkingParallelPair.one

Next tactic:
slice_lhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]