theorem pentagon_bimod {V W X Y Z : Mon_ C} (M : Bimod V W) (N : Bimod W X) (P : Bimod X Y)
    (Q : Bimod Y Z) :
    tensorHom (associatorBimod M N P).hom (𝟙 Q) ≫
      (associatorBimod M (N.tensorBimod P) Q).hom ≫
        tensorHom (𝟙 M) (associatorBimod N P Q).hom =
      (associatorBimod (M.tensorBimod N) P Q).hom ≫
        (associatorBimod M N (P.tensorBimod Q)).hom := by
  dsimp [tensorHom, associatorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  dsimp only [AssociatorBimod.hom]
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  slice_lhs 2 3 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.homAux]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp
  slice_lhs 1 2 => rw [← comp_tensor_id, coequalizer.π_desc]
  slice_rhs 1 3 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_rhs 3 4 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp
  slice_lhs 1 2 =>
    rw [← comp_tensor_id, π_tensor_id_preserves_coequalizer_inv_desc, comp_tensor_id,
      comp_tensor_id]
  slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  dsimp only [TensorBimod.X]
  slice_lhs 2 3 => rw [associator_naturality]
  slice_lhs 5 6 => rw [ι_colimMap, parallelPairHom_app_one]
  

Proof state:
case h.h
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)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((α_ M.X N.X P.X).hom ⊗ 𝟙 Q.X) ≫
      (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
        (𝟙 M.X ⊗ coequalizer.π (N.actRight ⊗ 𝟙 P.X) ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft)) ⊗ 𝟙 Q.X) ≫
          (𝟙 M.X ⊗
              coequalizer.π (TensorBimod.actRight N P ⊗ 𝟙 Q.X)
                ((α_ (coequalizer (N.actRight ⊗ 𝟙 P.X) ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))) Y.X Q.X).hom ≫
                  (𝟙 (coequalizer (N.actRight ⊗ 𝟙 P.X) ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))) ⊗ Q.actLeft))) ≫
            (𝟙 M.X ⊗
                coequalizer.desc
                  ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ⊗ 𝟙 P.X)
                        ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))).inv ≫
                    coequalizer.desc
                      ((α_ N.X P.X Q.X).hom ≫
                        (𝟙 N.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                          coequalizer.π
                            (N.actRight ⊗
                              𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))))
                            ((α_ N.X X.X
                                  (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom ≫
                              (𝟙 N.X ⊗ TensorBimod.actLeft P Q)))
                      (_ :
                        (tensorRight Q.X).map (N.actRight ⊗ 𝟙 P.X) ≫
                            (α_ N.X P.X Q.X).hom ≫
                              (𝟙 N.X ⊗
                                  coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                  ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)) =
                          (tensorRight Q.X).map ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft)) ≫
                            (α_ N.X P.X Q.X).hom ≫
                              (𝟙 N.X ⊗
                                  coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                  ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
                  (_ :
                    ((tensorBimod N P).actRight ⊗ 𝟙 Q.X) ≫ AssociatorBimod.homAux N P Q =
                      ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft)) ≫
                        AssociatorBimod.homAux N P Q)) ≫
              colimit.ι
                (parallelPair
                  (M.actRight ⊗
                    𝟙
                      (coequalizer (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
                  ((α_ M.X W.X
                        (coequalizer (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                          ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)))).hom ≫
                    (𝟙 M.X ⊗ TensorBimod.actLeft N (tensorBimod P Q))))
                WalkingParallelPair.one =
    ((coequalizer.π (M.actRight ⊗ 𝟙 N.X) ((α_ M.X W.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ⊗ 𝟙 P.X) ⊗ 𝟙 Q.X) ≫
      (α_ (coequalizer (M.actRight ⊗ 𝟙 N.X) ((α_ M.X W.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft))) P.X Q.X).hom ≫
        (𝟙 (coequalizer (M.actRight ⊗ 𝟙 N.X) ((α_ M.X W.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft))) ⊗
            coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
          (PreservesCoequalizer.iso
                (tensorRight (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))))
                (M.actRight ⊗ 𝟙 N.X) ((α_ M.X W.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft))).inv ≫
            coequalizer.desc
              ((α_ M.X N.X (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom ≫
                (𝟙 M.X ⊗
                    coequalizer.π
                      (N.actRight ⊗ 𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))))
                      ((α_ N.X X.X
                            (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom ≫
                        (𝟙 N.X ⊗ TensorBimod.actLeft P Q))) ≫
                  coequalizer.π
                    (M.actRight ⊗
                      𝟙
                        (coequalizer (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                          ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
                    ((α_ M.X W.X
                          (coequalizer (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                            ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)))).hom ≫
                      (𝟙 M.X ⊗ TensorBimod.actLeft N (tensorBimod P Q))))
              (_ :
                (tensorRight (tensorBimod P Q).X).map (M.actRight ⊗ 𝟙 N.X) ≫
                    (α_ M.X N.X (tensorBimod P Q).X).hom ≫
                      (𝟙 M.X ⊗
                          coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                            ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))) ≫
                        coequalizer.π (M.actRight ⊗ 𝟙 (tensorBimod N (tensorBimod P Q)).X)
                          ((α_ M.X W.X (tensorBimod N (tensorBimod P Q)).X).hom ≫
                            (𝟙 M.X ⊗ (tensorBimod N (tensorBimod P Q)).actLeft)) =
                  (tensorRight (tensorBimod P Q).X).map ((α_ M.X W.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ≫
                    (α_ M.X N.X (tensorBimod P Q).X).hom ≫
                      (𝟙 M.X ⊗
                          coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                            ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))) ≫
                        coequalizer.π (M.actRight ⊗ 𝟙 (tensorBimod N (tensorBimod P Q)).X)
                          ((α_ M.X W.X (tensorBimod N (tensorBimod P Q)).X).hom ≫
                            (𝟙 M.X ⊗ (tensorBimod N (tensorBimod P Q)).actLeft)))

Next tactic:
slice_lhs 4 5 => rw [← id_tensor_comp, coequalizer.π_desc]