-- 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
⊢ coequalizer.π ((M.tensorBimod (regular Y)).actRight ▷ N.X)
        (Iso.hom (α_ (M.tensorBimod (regular Y)).X Y.X N.X) ≫ (M.tensorBimod (regular Y)).X ◁ N.actLeft) ≫
      Hom.hom
        (Iso.hom
            (isoOfIso
              { hom := AssociatorBimod.hom M (regular Y) N, inv := AssociatorBimod.inv M (regular Y) N, hom_inv_id := ⋯,
                inv_hom_id := ⋯ :
                TensorBimod.X (M.tensorBimod (regular Y)) N ≅ TensorBimod.X M ((regular Y).tensorBimod N) }
              ⋯ ⋯) ≫
          M.whiskerLeft
            (Iso.hom
              (isoOfIso
                { hom := LeftUnitorBimod.hom N, inv := LeftUnitorBimod.inv N, hom_inv_id := ⋯, inv_hom_id := ⋯ :
                  TensorBimod.X (regular Y) N ≅ N.X }
                ⋯ ⋯))) =
    coequalizer.π ((M.tensorBimod (regular Y)).actRight ▷ N.X)
        (Iso.hom (α_ (M.tensorBimod (regular Y)).X Y.X N.X) ≫ (M.tensorBimod (regular Y)).X ◁ N.actLeft) ≫
      Hom.hom
        (whiskerRight
          (Iso.hom
            (isoOfIso
              { hom := RightUnitorBimod.hom M, inv := RightUnitorBimod.inv M, hom_inv_id := ⋯, inv_hom_id := ⋯ :
                TensorBimod.X M (regular Y) ≅ M.X }
              ⋯ ⋯))
          N)
have h : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶ TensorBimod.X M N)
  (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) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (regular Y)) N.X ⟶
        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))
    (AssociatorBimod.hom M (regular Y) 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) :
      TensorBimod.X (M.tensorBimod (regular Y)) N ⟶ TensorBimod.X M N))
  (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) sorry sorry)))
Tactic: dsimp