-- 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 :=
      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) ⋯ ⋯) :
        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) ⋯ ⋯)))
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))
    (coequalizer.desc (AssociatorBimod.homAux M (regular Y) N) sorry ≫
      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)))
  (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 [AssociatorBimod.hom]