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]
  slice_lhs 3 4 => 

Proof state:
case a.a
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
| 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)))
case a
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
case a.a
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
| 𝟙 M.X ⊗ coequalizer.π (Y.mul ⊗ 𝟙 N.X) ((α_ Y.X Y.X N.X).hom ≫ (𝟙 Y.X ⊗ N.actLeft))

Next tactic:
rw [ι_colimMap, parallelPairHom_app_one]