Let $\mathcal{C}$ be a category and $\mathcal{P}=Functors(\mathcal{C}^{op},\mathcal{S})$ the category of simplicial presheaves, where $\mathcal{S}=sSet$. I want $\mathcal{P}$ to be enriched, tensored, and cotensored over the category of simplicial sets $\mathcal{S}$ in the correct way. So I'll spell this out myself, and then the question will be if I got it right. (I'm pretty sure the answer is yes, but I'm uncertain. A reference where this is spelled out would be very welcome.)

First, to any simplicial set $K \in \mathcal{S}$, I can associate a constant simplicial presheaf, also denoted $K$. Then I can define an action of the symmetrical monoidal category $sSet$ on $\mathcal{P}$ by taking the (categorical) product with $K$ (which is computed levelwise). This, I think, is going to be the tensoring.

Next, for any two $F,G \in \mathcal{P}$, we want mapping spaces $Map_{\mathcal{P}}(F,G) \in \mathcal{S}$ so that the $0$-simplices are just the morphisms between $F,G$ in $\mathcal{P}$, and so that this is compatible with the tensoring: $Map_{\mathcal{P}}(K \times F,G) \simeq Map_{\mathcal{S}}(K, Map_{\mathcal{P}}(F,G))$. In particular, setting $K=\Delta^{n}$, we see that we must have for the $n$-simplices $Map_{\mathcal{P}}(F,G)_{n}=Hom_{\mathcal{P}}(\Delta^{n} \times F,G)$.

To the categorical product with constant simplicial presheaves and the above mapping spaces should make $\mathcal{P}$ tensored and enriched over $\mathcal{S}$.

Finally, we want $\mathcal{P}$ to be 'cotensored' or 'powered'. In fact, I think more is true. $\mathcal{P}$ should have an internal Hom whose value at $x \in \mathcal{C}$ is $\mathcal{Hom}(F,G)(x)=Map_{\mathcal{P}}(F_{| \mathcal{C}/x},G_{| \mathcal{C}/x})$, and the cotensoring $F^{K}$ should just be $\mathcal{Hom}_{\mathcal{P}}(K,G)$.