Given a base site $S$ of schemes, consider a stack $\mathcal{C}$ on $S$ not assumed to be algebraic. Then in principle, given another stack $\mathcal{E}$ on $S$, together with a map of stacks $\mathcal{E} \to \mathcal{C}$ over $S$, we could ask for $\mathcal{E}$ to be *algebraic over $\mathcal{C}$*. This would entail, aside from having an appropriate surjective map from a scheme or algebraic space, demanding that $\mathcal{E} \to \mathcal{E} \times_{\mathcal{C}} \mathcal{E}$ was representable. At this point I'm being agnostic whether $\mathcal{E}$ is in fact a stack over $\mathcal{C}$, since the definition I give doesn't require this (and if $\mathcal{C}$ is not fibred in groupoids, which is conceivable, this is not always possible to organise).

Has this concept appeared in the literature? I cannot find anything about it in the Stacks Project.

**EDIT** In the comments Ariyan Javanpeykar identifies this concept as a map of stacks being *representable by algebraic stacks*, which appears a few times in the literature without being formally defined and I guess must be a kind of folklore definition (it's not too difficult to guess, but I cannot find a paper that uses the phrase that also gives a definition). What I would like to use such a thing for is to look at geometric properties of maps of stacks that are representable by algebraic stacks, much as one can talk about geometric properties of maps that are representable (by schemes, or by algebraic spaces, even). It may be one gains nothing new, but it might allow for a more flexible definition in some cases.