Alper in his textbook Stacks and moduli defines the prestack of coherent sheaves over a smooth projective curve:
Let C be a fixed smooth, connected, and projective curve over an algebraically closed field $\mathbb k$. We define the prestack $Coh(C)$ over $Sch/\mathbb k$ where objects are pairs $(E, S)$ where $S$ is a scheme over $\mathbb k$ and $E$ is a coherent sheaf on $C_S = > C \times_{\mathbb k} S$ flat over $S$.
I struggle to see why smooth projective curves are somehow distinguished there, as the construction doesn't seem to use the fact that $C$ is a curve anywhere, so it should make sense also for more general schemes. Is there some reason to restrict attention to this specific type of base scheme that I've missed?