-- 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 M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
        colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one :
      (parallelPair (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft)).obj WalkingParallelPair.one ⟶
        colimit (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)))
    (CategoryStruct.comp (Y := coequalizer (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft))
        (coequalizer.π (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶ coequalizer (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft))
        (LeftUnitorBimod.hom M ≫ f.hom ≫ LeftUnitorBimod.inv N :
          TensorBimod.X
              { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }
              M ⟶
            TensorBimod.X
              { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }
              N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
        TensorBimod.X
          { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }
          N)
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (CategoryStruct.comp (Y := coequalizer (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft))
        (coequalizer.π (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft))
        (colimMap
          (parallelPairHom (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft) (μ ▷ N.X)
            (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) X.X X.X ◁ f.hom) (X.X ◁ f.hom)
            ⋯ ⋯)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
        TensorBimod.X
          { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }
          N)
    (CategoryStruct.comp (Y := coequalizer (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft))
        (coequalizer.π (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶ coequalizer (μ ▷ M.X) (Iso.hom (α_ X.X X.X M.X) ≫ X.X ◁ M.actLeft))
        (LeftUnitorBimod.hom M ≫ f.hom ≫ LeftUnitorBimod.inv N :
          TensorBimod.X
              { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }
              M ⟶
            TensorBimod.X
              { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }
              N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
        TensorBimod.X
          { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod X X }
          N)

Tactic:
slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.Limits.parallelPairHom_app_one