-- 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 Z : Mon.{v₁, u₁} C
M : Bimod X Y
N : Bimod Y Z
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶ TensorBimod.X M N)
    (CategoryStruct.comp (Y := TensorBimod.X M ((regular Y).tensorBimod N))
      (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) Y.X)
              N.X)
            (M.actRight ▷ Y.X ▷ N.X) ((Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ) ▷ N.X))
          (Iso.inv (PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) (Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ)))
          (coequalizer.desc
            (Iso.hom (α_ M.X Y.X N.X) ≫
              CategoryStruct.comp (Y :=
                tensorObj.{v₁, u₁} (𝒞 := inst) M.X (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
                (M.X ◁ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) Y.X N.X) ⟶
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
                (coequalizer.π (M.actRight ▷ TensorBimod.X (regular Y) N)
                    (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)) ⟶
                    coequalizer (X :=
                      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X)
                        (TensorBimod.X (regular Y) N))
                      (M.actRight ▷ TensorBimod.X (regular Y) N)
                      (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)))
            ⋯) :
        tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶ TensorBimod.X M ((regular Y).tensorBimod N))
      (colimMap
          (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N)
            (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)
            (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft)
            (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) :
        colimit
            (parallelPair (X :=
              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) (TensorBimod.X (regular Y) N))
              (M.actRight ▷ TensorBimod.X (regular Y) N)
              (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)) ⟶
          colimit (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))))
    (CategoryStruct.comp (Y :=
      coequalizer (TensorBimod.actRight M (regular Y) ▷ N.X)
        (Iso.hom (α_ (TensorBimod.X M (regular Y)) Y.X N.X) ≫ TensorBimod.X M (regular Y) ◁ N.actLeft))
      (coequalizer.π (TensorBimod.actRight M (regular Y) ▷ N.X)
        (Iso.hom (α_ (TensorBimod.X M (regular Y)) Y.X N.X) ≫ TensorBimod.X M (regular Y) ◁ N.actLeft))
      (colimMap
        (parallelPairHom (TensorBimod.actRight M (regular Y) ▷ N.X)
          (Iso.hom (α_ (TensorBimod.X M (regular Y)) Y.X N.X) ≫ TensorBimod.X M (regular Y) ◁ N.actLeft)
          (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft) (RightUnitorBimod.hom M ▷ Y.X ▷ N.X)
          (RightUnitorBimod.hom M ▷ N.X) ⋯ ⋯)))
have h : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶ TensorBimod.X M N)
  (CategoryStruct.comp (Y := TensorBimod.X M ((regular Y).tensorBimod N))
      (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) Y.X)
              N.X)
            (M.actRight ▷ Y.X ▷ N.X) ((Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ) ▷ N.X))
          (Iso.inv (PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) (Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ)))
          (coequalizer.desc
            (Iso.hom (α_ M.X Y.X N.X) ≫
              CategoryStruct.comp (Y :=
                tensorObj.{v₁, u₁} (𝒞 := inst) M.X (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
                (M.X ◁ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) Y.X N.X) ⟶
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
                (coequalizer.π (M.actRight ▷ TensorBimod.X (regular Y) N)
                    (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)) ⟶
                    coequalizer (X :=
                      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X)
                        (TensorBimod.X (regular Y) N))
                      (M.actRight ▷ TensorBimod.X (regular Y) N)
                      (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)))
            sorry) :
        tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶ TensorBimod.X M ((regular Y).tensorBimod N))
      (colimMap
          (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N)
            (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)
            (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft)
            (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) sorry sorry) :
        colimit
            (parallelPair (X :=
              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) (TensorBimod.X (regular Y) N))
              (M.actRight ▷ TensorBimod.X (regular Y) N)
              (Iso.hom (α_ M.X Y.X (TensorBimod.X (regular Y) N)) ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)) ⟶
          colimit (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶ TensorBimod.X M N)
  (RightUnitorBimod.hom M ▷ N.X ≫
      colimit.ι (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one :
    (parallelPair (TensorBimod.actRight M (regular Y) ▷ N.X)
            (Iso.hom (α_ (TensorBimod.X M (regular Y)) Y.X N.X) ≫ TensorBimod.X M (regular Y) ◁ N.actLeft)).obj
        WalkingParallelPair.one ⟶
      colimit (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft)))
Tactic: slice_rhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]