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 $S$)?