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 => 

Proof state:
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 (TensorBimod.actRight M (regular Y) ⊗ 𝟙 N.X)
      ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ (𝟙 (TensorBimod.X M (regular Y)) ⊗ N.actLeft)) ≫
    colimMap
      (parallelPairHom (TensorBimod.actRight M (regular Y) ⊗ 𝟙 N.X)
        ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ (𝟙 (TensorBimod.X M (regular Y)) ⊗ N.actLeft))
        (M.actRight ⊗ 𝟙 N.X) ((α_ M.X Y.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ((RightUnitorBimod.hom M ⊗ 𝟙 Y.X) ⊗ 𝟙 N.X)
        (RightUnitorBimod.hom M ⊗ 𝟙 N.X)
        (_ :
          ((tensorBimod M (regular Y)).actRight ⊗ 𝟙 N.X) ≫
              ((isoOfIso (Iso.mk (RightUnitorBimod.hom M) (RightUnitorBimod.inv M))
                      (_ :
                        (tensorBimod M (regular Y)).actLeft ≫ RightUnitorBimod.hom M =
                          (𝟙 X.X ⊗ RightUnitorBimod.hom M) ≫ M.actLeft)
                      (_ :
                        (tensorBimod M (regular Y)).actRight ≫ RightUnitorBimod.hom M =
                          (RightUnitorBimod.hom M ⊗ 𝟙 Y.X) ≫ M.actRight)).hom.hom ⊗
                (𝟙 N).hom) =
            (((isoOfIso (Iso.mk (RightUnitorBimod.hom M) (RightUnitorBimod.inv M))
                        (_ :
                          (tensorBimod M (regular Y)).actLeft ≫ RightUnitorBimod.hom M =
                            (𝟙 X.X ⊗ RightUnitorBimod.hom M) ≫ M.actLeft)
                        (_ :
                          (tensorBimod M (regular Y)).actRight ≫ RightUnitorBimod.hom M =
                            (RightUnitorBimod.hom M ⊗ 𝟙 Y.X) ≫ M.actRight)).hom.hom ⊗
                  𝟙 Y.X) ⊗
                (𝟙 N).hom) ≫
              (M.actRight ⊗ 𝟙 N.X))
        (_ :
          ((α_ (tensorBimod M (regular Y)).X Y.X N.X).hom ≫ (𝟙 (tensorBimod M (regular Y)).X ⊗ N.actLeft)) ≫
              ((isoOfIso (Iso.mk (RightUnitorBimod.hom M) (RightUnitorBimod.inv M))
                      (_ :
                        (tensorBimod M (regular Y)).actLeft ≫ RightUnitorBimod.hom M =
                          (𝟙 X.X ⊗ RightUnitorBimod.hom M) ≫ M.actLeft)
                      (_ :
                        (tensorBimod M (regular Y)).actRight ≫ RightUnitorBimod.hom M =
                          (RightUnitorBimod.hom M ⊗ 𝟙 Y.X) ≫ M.actRight)).hom.hom ⊗
                (𝟙 N).hom) =
            (((isoOfIso (Iso.mk (RightUnitorBimod.hom M) (RightUnitorBimod.inv M))
                        (_ :
                          (tensorBimod M (regular Y)).actLeft ≫ RightUnitorBimod.hom M =
                            (𝟙 X.X ⊗ RightUnitorBimod.hom M) ≫ M.actLeft)
                        (_ :
                          (tensorBimod M (regular Y)).actRight ≫ RightUnitorBimod.hom M =
                            (RightUnitorBimod.hom M ⊗ 𝟙 Y.X) ≫ M.actRight)).hom.hom ⊗
                  𝟙 Y.X) ⊗
                (𝟙 N).hom) ≫
              (α_ M.X Y.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)))

Next tactic:
rw [ι_colimMap, parallelPairHom_app_one]