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