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 => 

Proof state:
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
| 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)) ≫
    coequalizer.desc (AssociatorBimod.homAux M (regular Y) N)
      (_ :
        ((tensorBimod M (regular Y)).actRight ⊗ 𝟙 N.X) ≫ AssociatorBimod.homAux M (regular Y) N =
          ((α_ (tensorBimod M (regular Y)).X Y.X N.X).hom ≫ (𝟙 (tensorBimod M (regular Y)).X ⊗ N.actLeft)) ≫
            AssociatorBimod.homAux M (regular Y) N)
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
| colimMap
    (parallelPairHom (M.actRight ⊗ 𝟙 (TensorBimod.X (regular Y) N))
      ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ (𝟙 M.X ⊗ TensorBimod.actLeft (regular Y) 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 (regular Y) 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 (regular Y) N).X).hom ≫ (𝟙 M.X ⊗ (tensorBimod (regular Y) 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)))

Next tactic:
rw [coequalizer.π_desc]