-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
refine :
  CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft R.X)
        (coequalizer
          ({ X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.actRight ▷
            P.X)
          (Iso.hom
              (α_
                { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.X
                R.X P.X) ≫
            { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.X ◁
              P.actLeft)))
      (CategoryTheory.Functor.map (tensorLeft R.X)
        (coequalizer.π
          ({ X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.actRight ▷
            P.X)
          (Iso.hom
              (α_
                { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.X
                R.X P.X) ≫
            { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.X ◁
              P.actLeft)))
      (CategoryStruct.comp (Y :=
        TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) R.X
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
            (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
          (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
          (colimMap
            (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
              (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
              (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
                Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
              (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ⋯ ⋯)))
        (coequalizer.desc P.actLeft ⋯)) =
    CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft R.X)
        (coequalizer
          ({ X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.actRight ▷
            P.X)
          (Iso.hom
              (α_
                { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.X
                R.X P.X) ≫
            { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.X ◁
              P.actLeft)))
      (CategoryTheory.Functor.map (tensorLeft R.X)
        (coequalizer.π
          ({ X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
            P.X)
          (Iso.hom
              (α_
                { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.X
                R.X P.X) ≫
            { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }.X ◁
              P.actLeft)))
      (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X
        (TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P) ⟶
      P.X)
    (CategoryStruct.comp (Y :=
      TensorBimod.X
        { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod R R }
        P)
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
          (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (colimMap
          (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
              Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
            (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ⋯ ⋯)))
      (coequalizer.desc P.actLeft ⋯))
    (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)

Tactic:
refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_

Premises used: CategoryTheory.Functor.map, CategoryTheory.MonoidalCategory.tensorLeft, Iff.mp, CategoryTheory.Limits.coequalizer.π, CategoryTheory.cancel_epi