-- 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
⊢ coequalizer.π (((M.tensorBimod N).tensorBimod P).actRight ▷ Q.X)
        (Iso.hom (α_ ((M.tensorBimod N).tensorBimod P).X Y.X Q.X) ≫ ((M.tensorBimod N).tensorBimod P).X ◁ Q.actLeft) ≫
      (whiskerRight
            (Iso.hom
              (isoOfIso
                { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ :
                  TensorBimod.X (M.tensorBimod N) P ≅ TensorBimod.X M (N.tensorBimod P) }
                ⋯ ⋯))
            Q ≫
          Iso.hom
              (isoOfIso
                { hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
                  hom_inv_id := ⋯, inv_hom_id := ⋯ :
                  TensorBimod.X (M.tensorBimod (N.tensorBimod P)) Q ≅
                    TensorBimod.X M ((N.tensorBimod P).tensorBimod Q) }
                ⋯ ⋯) ≫
            M.whiskerLeft
              (Iso.hom
                (isoOfIso
                  { hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯,
                    inv_hom_id := ⋯ : TensorBimod.X (N.tensorBimod P) Q ≅ TensorBimod.X N (P.tensorBimod Q) }
                  ⋯ ⋯))).hom =
    coequalizer.π (((M.tensorBimod N).tensorBimod P).actRight ▷ Q.X)
        (Iso.hom (α_ ((M.tensorBimod N).tensorBimod P).X Y.X Q.X) ≫ ((M.tensorBimod N).tensorBimod P).X ◁ Q.actLeft) ≫
      (Iso.hom
            (isoOfIso
              { hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
                hom_inv_id := ⋯, inv_hom_id := ⋯ :
                TensorBimod.X ((M.tensorBimod N).tensorBimod P) Q ≅ TensorBimod.X (M.tensorBimod N) (P.tensorBimod Q) }
              ⋯ ⋯) ≫
          Iso.hom
            (isoOfIso
              { hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
                hom_inv_id := ⋯, inv_hom_id := ⋯ :
                TensorBimod.X (M.tensorBimod N) (P.tensorBimod Q) ≅ TensorBimod.X M (N.tensorBimod (P.tensorBimod Q)) }
              ⋯ ⋯)).hom
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))
    (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) sorry sorry) ≫
      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) sorry
            sorry)) =
  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)
Tactic: dsimp