-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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)
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst)
        (TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        S.X ⟶
      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) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
            S.X)
          (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
        (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (colimMap
          (parallelPairHom (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
              Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
                R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
            (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ⋯ ⋯)))
      (coequalizer.desc P.actLeft ⋯))
    (coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight)
have refine : CategoryStruct.comp (Y :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight S.X)
      (coequalizer
        ({ X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
              right_assoc := sorry, middle_assoc := sorry : Bimod R R }.actRight ▷
          P.X)
        (Iso.hom
            (α_
              { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                  right_assoc := sorry, middle_assoc := sorry : Bimod R R }.X
              R.X P.X) ≫
          { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                right_assoc := sorry, middle_assoc := sorry : Bimod R R }.X ◁
            P.actLeft)))
    (CategoryTheory.Functor.map (tensorRight S.X)
      (coequalizer.π
        ({ X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
              right_assoc := sorry, middle_assoc := sorry : Bimod R R }.actRight ▷
          P.X)
        (Iso.hom
            (α_
              { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                  right_assoc := sorry, middle_assoc := sorry : Bimod R R }.X
              R.X P.X) ≫
          { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                right_assoc := sorry, middle_assoc := sorry : Bimod R R }.X ◁
            P.actLeft)))
    (CategoryStruct.comp (Y :=
      TensorBimod.X
        { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
          middle_assoc := sorry : Bimod R R }
        P)
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
            S.X)
          (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
        (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (colimMap
          (parallelPairHom (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
              Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
                R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
            (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) sorry sorry)))
      (coequalizer.desc P.actLeft sorry)) =
  CategoryStruct.comp (Y :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight S.X)
      (coequalizer
        ({ X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
              right_assoc := sorry, middle_assoc := sorry : Bimod R R }.actRight ▷
          P.X)
        (Iso.hom
            (α_
              { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                  right_assoc := sorry, middle_assoc := sorry : Bimod R R }.X
              R.X P.X) ≫
          { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                right_assoc := sorry, middle_assoc := sorry : Bimod R R }.X ◁
            P.actLeft)))
    (CategoryTheory.Functor.map (tensorRight S.X)
      (coequalizer.π
        ({ X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
              right_assoc := sorry, middle_assoc := sorry }.actRight ▷
          P.X)
        (Iso.hom
            (α_
              { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                  right_assoc := sorry, middle_assoc := sorry : Bimod R R }.X
              R.X P.X) ≫
          { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                right_assoc := sorry, middle_assoc := sorry : Bimod R R }.X ◁
            P.actLeft)))
    (coequalizer.desc P.actLeft sorry ▷ S.X ≫ P.actRight)
Tactic: refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_