-- 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 :
  (Iso.hom (α_ M.X N.X P.X) ▷ Q.X ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst)
          (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)))
          Q.X)
        ((M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) ▷ Q.X :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X))
              Q.X ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst)
              (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)))
              Q.X)
        (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) ▷
            Q.X :
          tensorObj.{v₁, u₁} (𝒞 := inst)
              (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)))
              Q.X ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (N.tensorBimod P)) Q.X) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) 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)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft))
          P.X)
        Q.X)
      (coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X)
      (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) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) P.X) Q.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (N.tensorBimod P)) Q.X)

Tactic:
rw [← comp_whiskerRight, π_tensor_id_preserves_coequalizer_inv_desc, comp_whiskerRight,
      comp_whiskerRight]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, π_tensor_id_preserves_coequalizer_inv_desc