theorem hom_right_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actRight ≫ hom P Q L =
      (hom P Q L ⊗ 𝟙 U.X) ≫ (P.tensorBimod (Q.tensorBimod L)).actRight := by
  dsimp; dsimp [hom, homAux]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  rw [tensorRight_map]
  slice_lhs 1 2 => rw [TensorBimod.π_tensor_id_actRight]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => rw [← comp_tensor_id, coequalizer.π_desc, comp_tensor_id]
  refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
  

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (tensorRight L.X ⋙ tensorRight U.X).map
        (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (α_ (tensorBimod P Q).X L.X U.X).hom ≫
        (𝟙 (tensorBimod P Q).X ⊗ L.actRight) ≫
          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ⊗ 𝟙 Q.X)
                ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))).inv ≫
            coequalizer.desc
              ((α_ P.X Q.X L.X).hom ≫
                (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
                  coequalizer.π (P.actRight ⊗ 𝟙 (TensorBimod.X Q L))
                    ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ (𝟙 P.X ⊗ TensorBimod.actLeft Q L)))
              (_ :
                (tensorRight L.X).map (P.actRight ⊗ 𝟙 Q.X) ≫
                    (α_ P.X Q.X L.X).hom ≫
                      (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
                        coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                          ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft)) =
                  (tensorRight L.X).map ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫
                    (α_ P.X Q.X L.X).hom ≫
                      (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
                        coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                          ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft))) =
    (tensorRight L.X ⋙ tensorRight U.X).map
        (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ⊗ 𝟙 Q.X)
                ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))).inv ⊗
            𝟙 U.X) ≫
          (coequalizer.desc
              ((α_ P.X Q.X L.X).hom ≫
                (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
                  coequalizer.π (P.actRight ⊗ 𝟙 (TensorBimod.X Q L))
                    ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ (𝟙 P.X ⊗ TensorBimod.actLeft Q L)))
              (_ :
                (tensorRight L.X).map (P.actRight ⊗ 𝟙 Q.X) ≫
                    (α_ P.X Q.X L.X).hom ≫
                      (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
                        coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                          ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft)) =
                  (tensorRight L.X).map ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫
                    (α_ P.X Q.X L.X).hom ≫
                      (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
                        coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                          ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft))) ⊗
            𝟙 U.X)) ≫
        TensorBimod.actRight P (tensorBimod Q L)

Next tactic:
dsimp