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

Tactic:
rw [π_tensor_id_preserves_coequalizer_inv_desc]

Premises used: π_tensor_id_preserves_coequalizer_inv_desc