proposition: Lemma tensor_of_comonoids_laws_assoc (mx my : comonoid M) : comonoid_laws_assoc M (tensor_of_comonoids_data mx my).
proof: Proof. unfold comonoid_laws_assoc. cbn. rewrite ! assoc'. rewrite (bifunctor_rightcomp M). etrans. { apply maponpaths. apply pathsinv0. rewrite ! assoc. do 2 apply maponpaths_2. apply pathsinv0. apply precompose_inner_swap_with_diag_on_left. } rewrite (bifunctor_leftcomp M). etrans. 2: { apply maponpaths. rewrite assoc. apply maponpaths_2. apply pathsinv0. apply precompose_inner_swap_with_diag_on_right. } rewrite ! assoc. rewrite <- ! tensor_comp_mor. etrans. 2: { do 3 apply maponpaths_2. apply comonoid_to_law_assoc. } etrans. 2: { do 2 apply maponpaths_2. apply maponpaths. apply comonoid_to_law_assoc. } rewrite ! tensor_comp_mor. rewrite ! assoc'. do 2 apply maponpaths. apply inner_swap_hexagon. Qed.
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