-- 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
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
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst)
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
            middle_assoc := ⋯ : Bimod X X }.X
        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)
    (coequalizer.π
        ({ X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
              right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
          M.X)
        (Iso.hom
            (α_
              { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }.X
              X.X M.X) ≫
          { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }.X ◁
            M.actLeft) ≫
      Hom.hom
        ({ X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
              right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }.whiskerLeft
          f))
    (CategoryStruct.comp (Y :=
      coequalizer
        ({ X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
              right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
          M.X)
        (Iso.hom
            (α_
              { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }.X
              X.X M.X) ≫
          { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }.X ◁
            M.actLeft))
      (coequalizer.π
        ({ X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
              right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
          M.X)
        (Iso.hom
            (α_
              { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }.X
              X.X M.X) ≫
          { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }.X ◁
            M.actLeft))
      (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 }
                ⋯ ⋯))))
⊢ 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 }
              ⋯ ⋯)))

Tactic:
apply coequalizer.hom_ext

Premises used: CategoryTheory.Limits.coequalizer.hom_ext