Let $\mathcal{C}$ be a small simplicial category and let $F\: : \:\mathcal{C}^{op}\to sSet$ be a simplicial functor, we denote with $$ \int_{\mathcal{C}}F $$ the category where objects are triples $\left( x,\left[n \right],C \right)$ such that $x\in F_{n}\left(C \right) $ and maps $$ \left(\phi, f \right)\: : \: \left( y,\left[m \right],C' \right)\to\left( x,\left[n \right],C \right) $$ where $\phi\in \operatorname{Hom}_{\Delta}\left(\left[ n\right], \left[m \right] \right) $ and $f\in \operatorname{Hom}_{\mathcal{C}}\left(C, C' \right) $ such that $$ F\left(f \right)_{n}\circ \phi\left(y \right)=x. $$ There is a natural functor $$ \pi_{F}\: : \: \int_{\mathcal{C}}F\to \mathcal{C}\times \Delta $$ My goal is try to understand the proof of

Theorem

Let $\mathcal{C},\mathcal{E}$ be simplicial tensored and cotensored categories (tensored and cotensored over finite simplicial sets). Assume that $\mathcal{E}$ is cocomplete and $\mathcal{C}$ is small. Let $sPh\left( \mathcal{C}\right)$ be the category of simplicial presheaves over $\mathcal{C}$ (where each functors is a simplicial functor). Let $H\: : \: \mathcal{C}\to \mathcal{E}$ be a simplicial functor. Then the simplicial functor \begin{eqnarray*} R\: : \: \mathcal{E}&\to& sPh\left( \mathcal{C}\right)\\ E&\to& R\left( E\right), \end{eqnarray*} where $R\left( E\right)\left(C \right):=\operatorname{Map}_{\mathcal{E}}\left(G \left(C \right), E \right)$ has a left adjoint \begin{eqnarray*} L\: : \: sPh\left( \mathcal{C}\right)&\to& \mathcal{E}\\ \end{eqnarray*} given by $$ L\left( F\right):=\operatorname{colim}_{\int_{\mathcal{C}}F}\left(H\left(-\right) \otimes \Delta\left[- \right]\circ \pi_{F}\left(x, \left[n\right],C \right) \right) $$ Moreover we have that the functor $L$ preserves tensor objects and the adjunction is simplicially enriched, i.e. $$ \operatorname{Map}_{\mathcal{E}}\left(L\left(F \right) ,E \right)\cong \operatorname{Map}_{sPh\left( \mathcal{C}\right)}\left(F,R\left( E\right) \right). $$ This is theorem 2.3.21 of (Franz Vogler, Derived Manifolds from Functors of Points, logos Verlag Berlin). I understand why there is an ordinary adjunction but I have some difficult to understand why it is simplicial. Could you help me please?