(May be a poor title, happy to update)

Recall that for a stack $\mathcal{X} \to Sch$ on schemes (e.g. fppf site) and a pair of morphisms $x,y\colon U\to \mathcal{X}$ ($U$ a representable stack) there is a sheaf $Hom_{\mathcal{X}_U}(x,y)$ on $Sch/U$. Call this a 'local hom-sheaf'.

Reading through Laumon & Moret-Bailly I see that the local hom-sheaf of a pair of coherent sheaves on a projective scheme $X$ is an algebraic space (and is in addition 'sort of affine', namely it arises from a coherent sheaf). The stack (in *categories*) $Coh_X$ is clearly not algebraic. My question is whether this sort of behaviour (representability of local hom-sheaves by algebraic spaces) happens for other stacks of categories, in particular for stacks whose underlying stacks of groupoids are moduli stacks like $\mathcal{M}_g,\mathcal{M}_{g,n}, \mathcal{A}_g, \mathcal{A}_{g,d}$ and so on, where we take not just isomorphisms between curves, abelian varieties etc in the same fibre, but more general morphisms as well.