Let $\operatorname{sSet}$ be the category of simplicial sets and $\operatorname{Top}$ the category of (say, compactly generated) topological spaces. There is a pair of adjoint functors called geometric realization and singular set:
$$ |\cdot |:\operatorname{sSet}\to \operatorname{Top}:S_\bullet $$
I am aware of some of the basic properties of this, but I am confused about the way it acts on function complexes, as I will explain.
Given two simplicail sets $X,Y$ we can form the simplicial set of maps between them, which we call the function complex and it is defined by $$ \operatorname{Map}(X,Y)_n = \operatorname{Hom}(\Delta^n \times X,Y) $$
with the obvious face/degeneracy maps. This serves as an "internal hom" functor and it turns $\operatorname{sSet}$ into a cartesian closed category. In $\operatorname{Top}$ we also have an internal hom by endowing the set of maps between two topological spaces with the compact open topology. The general rather vague question is
What is the relationship between the space $|\operatorname{Map}(X,Y)|$ and $\operatorname{Map}(|X|,|Y|)$?
If I didn't make a mistake, I have the following. There is always a canonical map $$ \phi:|\operatorname{Map}(X,Y)|\to \operatorname{Map}(|X|,|Y|) $$ which is adjoint to a map$$ \psi:\operatorname{Map}(X,Y)\to S_{\bullet}(\operatorname{Map}(|X|,|Y|)) $$
and we have a natural isomorphism $$ S_{\bullet}(\operatorname{Map}(|X|,|Y|))\simeq \operatorname{Map}(X,S_\bullet|Y|) $$
The composition with $\psi$ is the map induced from the unit of the adjunction $Y\to S_\bullet|Y|$ which is a weak equivalence. Hence, I suspect that if $Y$ is a Kan complex then $\psi$, and hence $\phi$, is a weak equivalnce, but I lack the precise argument.
I do not expect $\phi$ to be weak homotopy equivalence in general, but I would like to know what can be said about it in this generality. For example, is $\phi$ injective on $\pi_0$? is it a cofibration?
It all seems very standard and well known, but I find it difficult to extract the answers from the standard sources like Goerss-Jardine for example.