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
  

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
⊢ coequalizer.π ((tensorBimod M (regular Y)).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 (AssociatorBimod.hom M (regular Y) N) (AssociatorBimod.inv M (regular Y) N))
              (_ :
                (tensorBimod (tensorBimod M (regular Y)) N).actLeft ≫ AssociatorBimod.hom M (regular Y) N =
                  (𝟙 X.X ⊗ AssociatorBimod.hom M (regular Y) N) ≫ (tensorBimod M (tensorBimod (regular Y) N)).actLeft)
              (_ :
                (tensorBimod (tensorBimod M (regular Y)) N).actRight ≫ AssociatorBimod.hom M (regular Y) N =
                  (AssociatorBimod.hom M (regular Y) N ⊗ 𝟙 Z.X) ≫
                    (tensorBimod M (tensorBimod (regular Y) N)).actRight)).hom ≫
          Hom.mk
            (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))))).hom =
    coequalizer.π ((tensorBimod M (regular Y)).actRight ⊗ 𝟙 N.X)
        ((α_ (tensorBimod M (regular Y)).X Y.X N.X).hom ≫ (𝟙 (tensorBimod M (regular Y)).X ⊗ N.actLeft)) ≫
      (Hom.mk
          (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))))).hom

Next tactic:
dsimp