Let $X,Y$ be finite etale $T$ schemes for some scheme $T$ (assume the maps $X\rightarrow T,Y\rightarrow T$ are surjective). Then the sheaf $\mathcal{Hom}\_T(X,Y)$ on $\text{Sch}/T$ (with the etale topology) sending
$$(U\rightarrow T)\mapsto\text{Hom}\_U(X\_U,Y\_U)$$
is representable, since on each component of $T$ it is finite locally constant. Let $H\stackrel{f}{\rightarrow}T$ be the representing $T$-scheme.


This means that we have a natural isomorphism of functors $\mathcal{Hom}\_T(X,Y)\cong\text{Hom}\_T(\underline{\;\;},H)$, so that for any $T$-scheme $U$, $U$-valued points of $H$ are given the interpretation of morphisms $X\_U\rightarrow Y\_U$.


On the other hand, suppose $T\rightarrow S$ is a covering in the etale topology (not sure if this important), then you can consider $H$ as an $S$-scheme. If $V\rightarrow S$ is also an $S$-scheme, then is there a natural interpretation for the $V$-valued points of $H$ (ie morphisms $V\rightarrow H$ over $S$)?



If $T$ is an $S$ scheme, and $X,Y$ are $T$-schemes, then do the $S$-valued points of $\mathcal{Hom}_T(X,Y)$ have an interpretation?
This might be a disappointing kind of answer and maybe there are better ways, but what is straightforward is this: morphisms $V\to H$ over $S$ are in one-to-one correspondence with pairs $(p,\varphi)$ where $p:V\to T$ is a $V$-valued point of $T$ over the $V$-valued point $V\to S$ of $S$ and $\varphi:p^\*X\to p^\*Y$ is a morphism over $V$ between $p$-fibres of $X$ and $Y$. 


In other words, given $f:T\to S$ and $s:V\to S$,
$$
\mathrm{Hom}\_S(V,H)\approx\coprod\_{\substack{t:V\to T\\ft=s}}\mathrm{Hom}\_V(t^\*X,t^\*Y)
$$ 


