How does hocolim relate to Hom? In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like this:$\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\colim}{colim}\DeclareMathOperator{\hocolim}{hocolim}\DeclareMathOperator{\Ho}{Ho}\DeclareMathOperator{\holim}{holim}$ $$\Hom\nolimits_\mathcal{C}(\colim A_i,B)=\lim \Hom\nolimits_\mathcal{C}(A_i,B)$$
I am looking for a corresponding statement for hocolims - lets say in simplicial sets, but if there are more general statements, that's even better.
E.g. I could imagine
$$\Hom\nolimits_{\mathcal{C}}(\hocolim A_i,B)=\lim \Hom\nolimits_{\Ho(\mathcal{C})}(A_i,B)$$
- maybe one needs to have $B$ fibrant and the A_i cofibrant here, i.e. that the Homs on the right are $\mathbb{R}Homs$.
Using the internal Hom in simplicial sets I could also imagine versions like this:
$$\Hom(\colim A_i,B)=\holim \Hom(A_i,B)$$
$$\Hom(\colim A_i,B)=\holim \mathbb{R}\!\Hom(A_i,B)$$
What is the right statement and what is the place to learn this hocolim-yoga?
Thanks!
N.B.
 A: A formula like the one you are asking for could be the following (Bousfield-Kan, "Homotopy limits, completions and localizations", chapter XII, proposition 4.1):
$$
\mathrm{hom}_* (\mathrm{hocolim}\ \mathbf{A}, B) \cong \mathrm{holim}\ \mathrm{hom}_* (\mathbf{A}, B) \ .
$$
Here $B$ is a pointed simplicial set, $\mathbf{A} : I \longrightarrow \Delta^{\mathrm{o}}\mathbf{Set}_*$ a functor from a small category $I$ to the category of pointed simplicial sets, and for pointed simplicial sets $A, B$
$$
\mathrm{hom}_* (A,B) \in \Delta^{\mathrm{o}}\mathbf{Set}_*
$$
is the pointed simplicial function space which $n$-simplices are maps in $\Delta^{\mathrm{o}}\mathbf{Set}_* $
$$
\left( \Delta [n] \times A \right) / \left( \Delta [n] \times * \right) \longrightarrow B
$$
(op.cit., chapter VIII, 4.8).
I didn't go through the details, but, if it annoys you, it seems to me that you can drop the "pointed" thing everywhere just deleting "pointed", $*$, and $\Delta [n] \times * $ in what Bousfield-Kan say.
