For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y\_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$.


I denote by $\Delta$ the category of [simplices](http://ncatlab.org/nlab/show/simplex+category), by $\Omega$ the category of [symmetric dendroids](http://ncatlab.org/nlab/show/dendroidal+set#Trees) (or symmetric finite rooted trees) and by $\mathbf{Oper}$ the category of [symmetric operads](http://ncatlab.org/nlab/show/symmetric+operad). The last two categories were introduced by I. Moerdijk and I. Weiss in [WeissPhD](http://dspace.library.uu.nl/handle/1874/22859) and [MW07](http://arxiv.org/abs/math/0701293); in the same articles they introduced the notion of Boardman-Vogt tensor product of operads $\otimes\_{\text BV}\colon\mathbf{Oper}\times\mathbf{Oper}\to\mathbf{Oper}$, which endows the category of small operads with a symmetric closed monoidal structure.


As $\mathbf{Oper}$ is a Bénabou cosmos with the Boardman-Vogt tensor product, we can define a pair of adjoint functors $\tau\_d\dashv N\_d$ with the nerve-realization paradigm (using the fact the $\mathbf{Oper}$ is cocomplete) and a tensor product of dendroidal sets (*i.e.* presheaves on $\Omega$) $\otimes \colon\widehat \Omega\times \widehat\Omega\to \widehat\Omega$ defined as the left Kan extension of $N\_d(-\otimes -)\colon \Omega\times \Omega\to \widehat\Omega$ along $y\_\Omega$.


In §3.5 of the article [HHM15](http://arxiv.org/pdf/1305.3658v3.pdf) it is stated that the category of dendroidal sets $\widehat \Omega$ is *weakly enriched* over the cartesian closed category of simplical sets $\widehat \Delta$.


Thanks to the nerve-realization paradigm $\tau\_d\dashv N\_d$ (for the dendroidal-operadic case), it is enough to prove that there exists a natural isomorphism


$$
\varphi \colon y\_\Delta\!\cdot\otimes(y\_\Delta\!\cdot\otimes\, y\_\Omega\cdot) \to N\_d\bigl(\,\cdot\otimes\_{\text BV}(\,\cdot\otimes\_{\text BV}\cdot\,)\bigr):\Delta\times (\Delta\times \Omega) \to \widehat\Omega
$$
and hence the weak enrichement follows by astract non-sense and the associativity of the Boardman-Vogt tensor product.


More generally, fixed an object $R$ of $\Omega$, the functor $\Omega^R\otimes\cdot\,\colon \widehat\Omega \to \widehat\Omega$ can be express as the left Kan extension of $N\_d\circ R\otimes\_{\text BV}\cdot$ along $y\_\Omega$. Thus


$$
\bigl( \Delta^m\otimes (\Delta^n \otimes \Omega^T)\bigr)\_S
\cong \int^R \widehat \Omega(\Omega^R, \Delta^n \otimes \Omega^T)\times N\_d([m]\otimes\_{\text BV} R)\_S\\
\phantom{AALA\bigl( \Omega^R\otimes (\Omega^S \otimes \Omega^T)\bigr)\_Q}\cong \int^R \mathbf{Oper}(R, [n]\otimes\_{\text BV} T) \times \mathbf{Oper}(S, [m]\otimes\_{\text BV} R)\\
$$
for any $[m], [n]$ in $\Delta$, $T$ in $\text{Ob}\Omega$, and where we have denoted by $\Omega^T$ the presheaf $y\_\Omega T$ on $\Omega$. Amazingly enough,


$$
\int^{R\in \Omega} \mathbf{Oper}(R, [n]\otimes\_{\text BV} T) \times \mathbf{Oper}(S, [m]\otimes\_{\text BV} R) \cong \int^{\mathcal P \in \mathbf{Oper}} \mathbf{Oper}(\mathcal P, [n]\otimes\_{\text BV} T) \times \mathbf{Oper}(S, [m]\otimes\_{\text BV} \mathcal P)
$$
where the last term is exaclty the coend form ([Yoneda expansion](http://ncatlab.org/nlab/show/co-Yoneda+lemma#every_presheaf_is_a_colimit_of_representables)) of the functor $N\_d([m]\otimes\_{\text BV}\cdot\,)\_S\colon \mathbf{Oper} \to \mathbf{Set}$ evaluated in $[n]\otimes\_{\text BV} T$.


Using the adjunction $\tau\_d\dashv N\_d$ and the fact that $\tau\_d$ distributes over the tensor product of dendroidal sets, it is possible to get then a natural transformation
$$
 \alpha\colon (y\_\Delta\cdot\times y\_\Delta\cdot\,)\otimes y\_\Omega\cdot \to y\_\Delta\cdot\otimes (y\_\Delta\cdot\otimes y\_\Omega\cdot\,) : \Delta \times \Delta \times \Omega \to \widehat\Omega\ .
$$


**Question 1.** What is an example of a pair of operads $\mathcal{P, Q}$ in $\text{Ob}\mathbf{Oper}$ such that $N\_d(\mathcal P\otimes\_{\text BV}\mathcal Q)$ is *not* isomorphic to $N\_d(\mathcal P) \otimes N\_d(\mathcal Q)$, *i.e.* for which the nerve does not distribute over the Boardman-Vogt tensor product?


It is stated in §3.5 of [HHM15](http://arxiv.org/pdf/1305.3658v3.pdf) that the category $\widehat\Omega$ of dendroidal sets is not a monoidal category with the tensor product, as the associativity fails. Indeed, they prove a *weak* associativity for representable presheaves, as I worked out above, and then it is stated that extending by colimits to all triple of presheaves on $\Omega$ *that* particular natural isomorphism between triple of representable presheaves, then the comparison morphism is not an isomorphism in general.


**Question 2.** What is an example of a triple of presheaves $X, Y, Z$ on $\Omega$ such that the comparison morphism built in [HHM15](http://arxiv.org/pdf/1305.3658v3.pdf) is not an isomorphism? An example for which $X, Y$ are actually presheaves on $\Delta$ would be appreciated even more.


**Question 3.** Why cannot exist any other *associator* $\alpha$ (natural isomorphism for the associativity) making $(\widehat \Omega, \otimes, \alpha, \lambda, \rho, \underline{hom}\_\Omega)$ a closed symmetric monoidal category?


For any presheaves $X, Y$ on $\Omega$ and any object $T$ of $\Omega$, we define $\underline{hom}\_\Omega(X, Y)\_T = \widehat\Omega(\Omega^T\otimes X, Y)$.



Tensor product of dendroidal sets: counter-examples
Associativity of the tensor product in fact does not hold even for representable presheaves, which addresses your Questions 2 and 3. Proposition 3.6.9 and its proof (in HHM, as you cite) give a typical counterexample, worked out explicitly there.


As to Question 1: consider the associative operad $\mathbf{A}$. Tensoring that with itself gives the commutative operad (essentially by the Eckman-Hilton argument), but the tensor product $N\_d(\mathbf{A}) \otimes N\_d(\mathbf{A})$ is actually a model for the homotopy-coherent nerve of the $\mathbf{E}\_2$-operad.


