Here by "algebraic stack" I mean a stack in groupoids over $(\textbf{Sch}/S)_\text{etale}$ (for some scheme $S$) whose diagonal morphism is representable (by schemes), and which is covered by a surjective etale morphism from a scheme.

The question is, if $\mathcal{X}\rightarrow\mathcal{Y}$ is a morphism of stacks in groupoids which is representable finite etale, and and $\mathcal{Y}$ is an algebraic stack, then is $\mathcal{X}$ also an algebraic stack?

You can certainly pull back the scheme cover of $\mathcal{Y}$ to get a surjective etale scheme cover of $\mathcal{X}$, so really I suppose I'm asking if the diagonal morphism of $\mathcal{X}$ must be representable.

The stacks project has a similar result (Lemmas 67.15.3, 67.15.4), but their definition of an algebraic stack only requires the diagonal morphism to be representable by algebraic spaces, but their result seems to rely on the fact that a stack covered by an algebraic space is an algebraic space, which of course isn't true if you replace algebraic spaces by schemes.