I would like to prove the following fact, which I learned from a previous MO question.


Let $S\_\cdot,T\_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T\_\cdot$ is Kan. Then there is a weak equivalence
$$
|\underline{\mathbf{sSET}} (S\_\cdot,T\_\cdot)|\simeq \underline{\mathbf{TOP}} (|S\_\cdot|,|T\_\cdot|)
$$


Here is what I have so far: By a Quillen adjunction,
$$
\mathbf{TOP} (|\underline{\mathbf{sSET}} (S\_\cdot,T\_\cdot)|, \underline{\mathbf{TOP}} (|S\_\cdot|,|T\_\cdot|)) \cong \mathbf{sSET}(\underline{\mathbf{sSET}} (S\_\cdot,T\_\cdot),\textrm{Sing}\ \underline{\mathbf{TOP}} (|S\_\cdot|,|T\_\cdot|))
$$
So we need to find a weak equivalence in the second set. Notice
$$
\textrm{Sing}\ \underline{\mathbf{TOP}} (|S\_\cdot|,|T\_\cdot|))\_k \cong \mathbf{TOP}(\Delta^k\times |S\_\cdot|,|T\_\cdot|)
$$
And by the universal property of coends, there's a map (\*)
$$
|\Delta(k)|\rightarrow \Delta^k 
$$  

($\Delta(k)$ is the simplicial set corepresenting $[k]\in\mathbb{\Delta}$), so we get a map to $\mathbf{TOP}(|\Delta(k) \times S\_\cdot|,|T\_\cdot|)$, which is in turn in bijection with $\mathbf{sSET}(\Delta(k) \times S\_\cdot,\textrm{Sing} |T\_\cdot|)$, and that is the $k^{\textrm{th}}$ space of $\underline{\mathbf{sSET}}(S\_\cdot,\textrm{Sing}\ |T\_\cdot|)$. So unraveling, we need to find a weak equivalence
$$
\underline{\mathbf{sSET}} (S\_\cdot,T\_\cdot) \rightarrow \underline{\mathbf{sSET}} (S\_\cdot, \textrm{Sing} |T\_\cdot|)
$$
We do have the unit map $T\_\cdot\rightarrow \textrm{Sing} |T\_\cdot|$ of the adjunction in the target, and since all simplicial sets are cofibrant, the result would follow if 
this unit map is a trivial fibration when $T\_\cdot$ is fibrant (by compatibility of inner-hom with the model structure in $\mathbf{sSET}$). Here's where I'm stuck; it seems like I'm missing a key ingredient to finish.


(\*) also here I need to show that these set maps
$$
\mathbf{TOP}(\Delta^k\times |S\_\cdot|,|T\_\cdot|) \rightarrow \mathbf{TOP}(|\Delta(k) \times S\_\cdot|,|T\_\cdot|)
$$
assemble to a weak equivalence of simplicial sets.



Inner hom and geometric realization.
The unit map $T\to \mathrm{Sing}|T|$ is far from being a trivial fibration; it's actually injective. Did you mean to say it's a trivial *cofibration*?


It *is* a weak equivalence for any simplicial set $T$ (so the map is actually a trivial cofibration), and this would complete your proof. Unfortunately, this seems to take some hard work. Goerss & Jardine state this as Proposition 11.1 in chapter 1 of their book; their proof relies on much of the material in the previous 40 pages or so of the chapter. 


The main idea in their proof (which goes back to Quillen) is to show that geometric realization $|-|:\mathbf{sSet}\to \mathbf{Top}$ preserves fibrations. It's clear from the definitions that $\mathrm{Sing}$ preserves fibrations, so therefore the composite functor $\mathrm{Sing}|-|$ takes Kan fibrations to Kan fibrations. In particular, it preserves the path-loop fibration, so you can reduce the problem of comparing homotopy groups $\pi\_n(T,t)\to \pi\_n(\mathrm{S}|T|,t)$ to the case $n=0$.


Showing that $|-|$ preserves fibrations involves a detour through the theory of minimal Kan fibrations, which is charming; but it would be nice if there was a more direct proof. I don't know one.


