-- 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
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
dsimp :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X ◁
          coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ≫
        μ ▷ coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ≫ actLeft P Q :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
        coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X)
          (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
        (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X ◁
            coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X)
              (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X)
              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
        (Iso.hom (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
            CategoryStruct.comp (Y :=
              tensorObj.{v₁, u₁} (𝒞 := inst) R.X
                (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
              (R.X ◁ actLeft P Q :
                tensorObj.{v₁, u₁} (𝒞 := inst) R.X
                    (tensorObj.{v₁, u₁} (𝒞 := inst) R.X
                      (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))) ⟶
                  tensorObj.{v₁, u₁} (𝒞 := inst) R.X
                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
              (actLeft P Q : tensorObj.{v₁, u₁} (𝒞 := inst) R.X (X P Q) ⟶ X P Q) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X)
              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
        coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
⊢ Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X))
        (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
      X P Q)
    (CategoryTheory.Functor.map (tensorLeft (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X))
        (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ≫
      μ ▷ X P Q ≫ actLeft P Q)
    (CategoryTheory.Functor.map (tensorLeft (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X))
        (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ≫
      Iso.hom (α_ R.X R.X (X P Q)) ≫ R.X ◁ actLeft P Q ≫ actLeft P Q)

Tactic:
dsimp [X]

Premises used: Bimod.TensorBimod.X