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