Let $S,T$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. We let $|S|$ denote the geometric realisation of a simplicial set $S$.

Given $X$ and $Y$, (compactly generated) topological spaces, we use $TOP(X,Y)$ to denote the function space, with the k-ification of the compact open topology.

We also have a simplicial mapping space $TOP_{Simp}(X,Y)$, which is essentially the singular complex $Sing(TOP(X,Y))$ of $TOP(X,Y)$.

We have a weak homotopy equivalence $|Sing(X)| \to X$ given any topological space $X$.

There exists a well known weak homotopy equivalence $|SIMP(S,K)| \to TOP(|S|,|K|)$. It is essentially derived from the fact that $K$ is a strong deformation retract of the singular complex $Sing(|K|)$ if $K$ is Kan. This weak homotopy is the composition of the obvious maps: $$ |SIMP(S,K)| \to |SIMP(S,Sing|K|)| \stackrel{\cong}{\to} |TOP_{Simp}(|S|,|K|)|\to TOP(|S|,|K|). $$

Suppose that we have a cofibration (meaning inclusion) $i\colon T \to S$ of simplicial sets. We have an induced fibration $i^*\colon SIMP(S,K) \to SIMP(T,K)$ of simplicial sets, hence a Serre fibration $|i^*|\colon |SIMP(S,K)| \to |SIMP(T,K)|$.

We also have a cofibration $|i|\colon |T| \to |S|$, thus a (Hurewicz, hence Serre) fibration $|i|^*\colon TOP(|S|,|K|) \to TOP(|T|,|K|)$. (We just restrict a map $|S| \to |K|$ to $|T|\subset |S|$.)

It is also clear that we have a commutative diagram: $\require{AMScd}$ \begin{CD} |SIMP(S,K)| @>>> TOP(|S|,|K|)\\ @V |i^*| V V @VV |i|^* V\\ |SIMP(T,K)| @>>> TOP(|T|,|K|)\\ \end{CD}

The question I would like to ask is the following. Does $i^*\colon SIMP(S,K) \to SIMP(T,K)$ give a 'faithful' simplicial model of the fibration $|i|^*\colon TOP(|S|,|K|) \to TOP(|T|,|K|)$ (whatever this means). For instance (this would suffice), does the homotopy long exact sequence of $|i|^*$ coincide with that of $|i^*|$?