-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon.{v₁, u₁} C
M : Bimod X Y
N : Bimod Y Z
h :
  Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight N.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X (regular Y).X) ⟶
      TensorBimod.X M N)
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight N.X)
        (coequalizer (M.actRight ▷ (regular Y).X) (Iso.hom (α_ M.X Y.X (regular Y).X) ≫ M.X ◁ (regular Y).actLeft)))
      (CategoryTheory.Functor.map (tensorRight N.X)
        (coequalizer.π (M.actRight ▷ (regular Y).X) (Iso.hom (α_ M.X Y.X (regular Y).X) ≫ M.X ◁ (regular Y).actLeft)))
      (CategoryStruct.comp (Y := TensorBimod.X M ((regular Y).tensorBimod N))
        (CategoryStruct.comp (Y :=
            coequalizer (X :=
              tensorObj.{v₁, u₁} (𝒞 := inst)
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) Y.X) N.X)
              (M.actRight ▷ Y.X ▷ N.X) ((Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ) ▷ N.X))
            (Iso.inv
              (PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) (Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ)))
            (coequalizer.desc
              (Iso.hom (α_ M.X Y.X N.X) ≫
                CategoryStruct.comp (Y :=
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                    (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
                  (M.X ◁ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft) :
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) Y.X N.X) ⟶
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                        (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
                  (coequalizer.π (M.actRight ▷ TensorBimod.X (regular Y) N)
                      (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) :
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                        (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)) ⟶
                      coequalizer (X :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X)
                          (TensorBimod.X (regular Y) N))
                        (M.actRight ▷ TensorBimod.X (regular Y) N)
                        (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)))
              ⋯) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶
            TensorBimod.X M ((regular Y).tensorBimod N))
        (colimMap
            (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N)
              (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)
              (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft)
              (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) :
          colimit
              (parallelPair (X :=
                tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) (TensorBimod.X (regular Y) N))
                (M.actRight ▷ TensorBimod.X (regular Y) N)
                (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)) ⟶
            colimit (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft)))))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight N.X)
        (coequalizer (M.actRight ▷ (regular Y).X) (Iso.hom (α_ M.X Y.X (regular Y).X) ≫ M.X ◁ (regular Y).actLeft)))
      (CategoryTheory.Functor.map (tensorRight N.X)
        (coequalizer.π (M.actRight ▷ (regular Y).X) (Iso.hom (α_ M.X Y.X (regular Y).X) ≫ M.X ◁ (regular Y).actLeft)))
      (coequalizer.desc M.actRight ⋯ ▷ N.X ≫
        colimit.ι (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))
          WalkingParallelPair.one))
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶ TensorBimod.X M N)
    (CategoryStruct.comp (Y := TensorBimod.X M ((regular Y).tensorBimod N))
      (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) Y.X)
              N.X)
            (M.actRight ▷ Y.X ▷ N.X) ((Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ) ▷ N.X))
          (Iso.inv (PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) (Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ)))
          (coequalizer.desc
            (Iso.hom (α_ M.X Y.X N.X) ≫
              CategoryStruct.comp (Y :=
                tensorObj.{v₁, u₁} (𝒞 := inst) M.X (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
                (M.X ◁ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) Y.X N.X) ⟶
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
                (coequalizer.π (M.actRight ▷ TensorBimod.X (regular Y) N)
                    (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)) ⟶
                    coequalizer (X :=
                      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X)
                        (TensorBimod.X (regular Y) N))
                      (M.actRight ▷ TensorBimod.X (regular Y) N)
                      (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)))
            ⋯) :
        tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶ TensorBimod.X M ((regular Y).tensorBimod N))
      (colimMap
          (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N)
            (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)
            (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft)
            (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) :
        colimit
            (parallelPair (X :=
              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) (TensorBimod.X (regular Y) N))
              (M.actRight ▷ TensorBimod.X (regular Y) N)
              (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)) ⟶
          colimit (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))))
    (coequalizer.desc M.actRight ⋯ ▷ N.X ≫
      colimit.ι (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one)

Tactic:
refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_

Premises used: CategoryTheory.Functor.map, CategoryTheory.MonoidalCategory.tensorRight, Iff.mp, CategoryTheory.Limits.coequalizer.π, CategoryTheory.cancel_epi