-- 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)
V W X Y Z : Mon.{v₁, u₁} C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ CategoryStruct.comp (Y :=
      coequalizer (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
        (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft))
      (coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
        (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft))
      (colimMap
          (parallelPairHom (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
            (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft)
            (TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
            (Iso.hom (α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X) ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft)
            (AssociatorBimod.hom M N P ▷ Y.X ▷ Q.X) (AssociatorBimod.hom M N P ▷ Q.X) ⋯ ⋯) ≫
        AssociatorBimod.hom M (N.tensorBimod P) Q ≫
          colimMap
            (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
              (Iso.hom (α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)) ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
              (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
              (Iso.hom (α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))) ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
              (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X ◁ AssociatorBimod.hom N P Q) (M.X ◁ AssociatorBimod.hom N P Q) ⋯
              ⋯)) =
    coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
        (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
      AssociatorBimod.hom (M.tensorBimod N) P Q ≫ AssociatorBimod.hom M N (P.tensorBimod Q)
have h : CategoryStruct.comp (Y :=
    coequalizer (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
      (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft))
    (coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
      (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft))
    (CategoryStruct.comp (Y := TensorBimod.X (M.tensorBimod (N.tensorBimod P)) Q)
      (colimMap
        (parallelPairHom (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
          (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft)
          (TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
          (Iso.hom (α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X) ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft)
          (coequalizer.desc (AssociatorBimod.homAux M N P) sorry ▷ Y.X ▷ Q.X)
          (coequalizer.desc (AssociatorBimod.homAux M N P) sorry ▷ Q.X) sorry sorry))
      (coequalizer.desc (AssociatorBimod.homAux M (N.tensorBimod P) Q) sorry ≫
        colimMap
          (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
            (Iso.hom (α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)) ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
            (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
            (Iso.hom (α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))) ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
            (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) sorry)
            (M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) sorry) sorry sorry))) =
  CategoryStruct.comp (Y :=
    coequalizer (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
      (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft))
    (coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
      (Iso.hom (α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X) ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft))
    (coequalizer.desc (AssociatorBimod.homAux (M.tensorBimod N) P Q) sorry ≫
      coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) sorry)
Tactic: dsimp only [AssociatorBimod.hom]