Let $K/k$ be an arbitrary field extension and $X$, $Y$ varieties over $k$ (lets assume projective and perhaps smooth to avoid technicalities). There is a fine moduli space of morphisms between $X$ and $Y$ parametrized by a scheme $Hom(X,Y)$ over $k$. My question is whether $Hom(X,Y)\_K$ is isomorphic to $Hom(X\_K, Y\_K)$ where ${}\_K$ denotes tensoring with $Spec K$. From the universal property of $Hom(X,Y)$ we have a bijection between maps $Spec K\to Hom(X,Y)$ (i.e. $K$-rational points, as a set contained in $Hom(X,Y)\_K$) and maps $Y\_K\to X\_K$. I seem to be a bit confused as to whether this is enough to conclude.


Finally, could we get a similar thing to work for the coarse Kontsevich moduli space $\mathcal{M}\_{g,n}(X,\beta)$ even though there is no universal family? Thanks.



Fine/Coarse moduli spaces and extensions of fields. 
If $X,Y$ are $S$-schemes, then $\underline{\hom}\_S(X,Y)$ denotes the sheaf on $S$-schemes defined by $T \mapsto \hom\_T(X\_T,Y\_T)$. Now, if $S'/S$ is any base change, we have for every $S'$-scheme $T$:


$$\hom\_{S'}(T,\underline{\hom}\_S(X,Y)\_{S'}) = \hom\_S(T,\underline{\hom}\_S(X,Y)) = \hom\_T(X\_T,Y\_T)$$


$$ = \hom\_T((X\_{S'})\_T,(Y\_{S'})\_T)) = \hom\_{S'}(T,\underline{\hom}\_{S'}(X\_{S'},Y\_{S'}))$$


Hence, $\underline{\hom}\_S(X,Y)\_{S'} = \underline{\hom}\_{S'}(X\_{S'},Y\_{S'})$.


