Let $S$ be a base scheme. Let $(F_{i},f_{ii'})_{i\in I}$ be a directed inverse system of algebraic spaces over $S$. Then if each $f_{ii'}$ is affine, the inverse limit $\lim_{i}F_{i}$ exists as an algebraic space.
Let $({\cal{X}}_{i},g_{ii'})_{i\in I}$ be a directed inverse system of algebraic stacks over $S$. Then as the case of algebraic spaces, is the inverse limit $\lim_{i}\cal{X}_{i}$ exists as an algebraic stack? If not, when $\lim_{i}\cal{X}_{i}$ is an algebraic stack? In fact, it seems that there is no reference concerning the inverse limit $\lim_{i}\cal{X}_{i}$ of algebraic stacks. I have no idea what is the definition of the limit $\lim_{i}\cal{X}_{i}$. Is it simply a small limit in $Cat$?
This is what I found in Xinwen Zhu's paper (arXiv:1707.05700v1).
