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

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

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