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

Tactic:
rw [coequalizer.π_desc]

Premises used: CategoryTheory.Limits.coequalizer.π_desc