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.