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]
  

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
⊢ ((PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ⊗ 𝟙 Y.X) ((α_ M.X Y.X Y.X).hom ≫ (𝟙 M.X ⊗ Y.mul))).inv ≫
        coequalizer.desc
          ((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 (regular Y) N))
                ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ (𝟙 M.X ⊗ TensorBimod.actLeft (regular Y) N)))
          (_ :
            (tensorRight N.X).map (M.actRight ⊗ 𝟙 (regular Y).X) ≫
                (α_ M.X (regular Y).X N.X).hom ≫
                  (𝟙 M.X ⊗
                      coequalizer.π ((regular Y).actRight ⊗ 𝟙 N.X)
                        ((α_ (regular Y).X Y.X N.X).hom ≫ (𝟙 (regular Y).X ⊗ N.actLeft))) ≫
                    coequalizer.π (M.actRight ⊗ 𝟙 (tensorBimod (regular Y) N).X)
                      ((α_ M.X Y.X (tensorBimod (regular Y) N).X).hom ≫ (𝟙 M.X ⊗ (tensorBimod (regular Y) N).actLeft)) =
              (tensorRight N.X).map ((α_ M.X Y.X (regular Y).X).hom ≫ (𝟙 M.X ⊗ (regular Y).actLeft)) ≫
                (α_ M.X (regular Y).X N.X).hom ≫
                  (𝟙 M.X ⊗
                      coequalizer.π ((regular Y).actRight ⊗ 𝟙 N.X)
                        ((α_ (regular Y).X Y.X N.X).hom ≫ (𝟙 (regular Y).X ⊗ N.actLeft))) ≫
                    coequalizer.π (M.actRight ⊗ 𝟙 (tensorBimod (regular Y) N).X)
                      ((α_ M.X Y.X (tensorBimod (regular Y) N).X).hom ≫
                        (𝟙 M.X ⊗ (tensorBimod (regular Y) N).actLeft)))) ≫
      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))) =
    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:
slice_rhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]