-- 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)
V W X Y Z : Mon.{v₁, u₁} C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
a :
  (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) N.X)
            P.X)
          (M.actRight ▷ N.X ▷ P.X) ((Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X))
        (Iso.inv
          (PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft)))
        (coequalizer.desc
          (Iso.hom (α_ M.X N.X P.X) ≫
            CategoryStruct.comp (Y :=
              tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)))
              (M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) :
                tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) ⟶
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                    (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)))
              (coequalizer.π (M.actRight ▷ TensorBimod.X N P)
                  (Iso.hom (α_ M.X W.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P) :
                tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                    (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) ⟶
                  coequalizer (X :=
                    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) (TensorBimod.X N P))
                    (M.actRight ▷ TensorBimod.X N P)
                    (Iso.hom (α_ M.X W.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P)))
          ⋯) ▷
      Q.X :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) P.X) Q.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (N.tensorBimod P)) Q.X)
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst)
      (coequalizer (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N) X.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
      Q.X)
    (coequalizer.π (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N) X.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft) ▷
      Q.X)
    (coequalizer.desc
        (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) N.X)
              P.X)
            (M.actRight ▷ N.X ▷ P.X) ((Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X))
          (Iso.inv
            (PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
              (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft)))
          (coequalizer.desc
            (Iso.hom (α_ M.X N.X P.X) ≫
              CategoryStruct.comp (Y :=
                tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                  (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)))
                (M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) ⟶
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)))
                (coequalizer.π (M.actRight ▷ TensorBimod.X N P)
                    (Iso.hom (α_ M.X W.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) ⟶
                    coequalizer (X :=
                      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) (TensorBimod.X N P))
                      (M.actRight ▷ TensorBimod.X N P)
                      (Iso.hom (α_ M.X W.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P)))
            ⋯))
        ⋯ ▷
      Q.X)

Tactic:
rw [← comp_whiskerRight, coequalizer.π_desc]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.Limits.coequalizer.π_desc