-- 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
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)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    (M.tensorBimod
          { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod Y Y }).X ⟶
      (N.tensorBimod
          { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod Y Y }).X)
    (Hom.hom
      (whiskerRight f
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ }))
    (Hom.hom
      (Iso.hom
          (isoOfIso { hom := RightUnitorBimod.hom M, inv := RightUnitorBimod.inv M, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
            ⋯) ≫
        f ≫
          Iso.inv
            (isoOfIso
              { hom := RightUnitorBimod.hom N, inv := RightUnitorBimod.inv N, hom_inv_id := ⋯, inv_hom_id := ⋯ :
                TensorBimod.X N
                    { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                      right_assoc := ⋯, middle_assoc := ⋯ : Bimod Y Y } ≅
                  N.X }
              ⋯ ⋯)))
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X
      { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
          middle_assoc := sorry : Bimod Y Y }.X ⟶
    (N.tensorBimod
        { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
          middle_assoc := sorry : Bimod Y Y }).X)
  (coequalizer.π
      (M.actRight ▷
        { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
            middle_assoc := sorry : Bimod Y Y }.X)
      (Iso.hom
          (α_ M.X Y.X
            { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                right_assoc := sorry, middle_assoc := sorry : Bimod Y Y }.X) ≫
        M.X ◁
          { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
              right_assoc := sorry, middle_assoc := sorry }.actLeft) ≫
    Hom.hom
      (whiskerRight f
        { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
          middle_assoc := sorry : Bimod Y Y }))
  (CategoryStruct.comp (Y :=
    coequalizer
      (M.actRight ▷
        { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
            middle_assoc := sorry : Bimod Y Y }.X)
      (Iso.hom
          (α_ M.X Y.X
            { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                right_assoc := sorry, middle_assoc := sorry : Bimod Y Y }.X) ≫
        M.X ◁
          { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
              right_assoc := sorry, middle_assoc := sorry : Bimod Y Y }.actLeft))
    (coequalizer.π
      (M.actRight ▷
        { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
            middle_assoc := sorry : Bimod Y Y }.X)
      (Iso.hom
          (α_ M.X Y.X
            { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                right_assoc := sorry, middle_assoc := sorry : Bimod Y Y }.X) ≫
        M.X ◁
          { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
              right_assoc := sorry, middle_assoc := sorry }.actLeft))
    (Hom.hom
      (Iso.hom
          (isoOfIso { hom := RightUnitorBimod.hom M, inv := RightUnitorBimod.inv M, hom_inv_id := sorry, inv_hom_id := sorry } sorry
            sorry) ≫
        f ≫
          Iso.inv
            (isoOfIso
              { hom := RightUnitorBimod.hom N, inv := RightUnitorBimod.inv N, hom_inv_id := sorry, inv_hom_id := sorry :
                TensorBimod.X N
                    { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry,
                      right_assoc := sorry, middle_assoc := sorry : Bimod Y Y } ≅
                  N.X }
              sorry sorry))))
Tactic: apply coequalizer.hom_ext