The Stacks project

Proof. Recall that by Groupoids in Spaces, Definition 78.18.1 (see also discussion following the definition) we have $R_ Z = s^{-1}(Z) = t^{-1}(Z)$ as locally closed subspaces of $R$. Hence the two morphisms $R_ Z \to Z$ are smooth as base changes of $s$ and $t$. Hence $(Z, R_ Z, s|_{R_ Z}, t|_{R_ Z}, c|_{R_ Z \times _{s, Z, t} R_ Z})$ is a smooth groupoid in algebraic spaces, and we see that $[Z/R_ Z]$ is an algebraic stack, see Algebraic Stacks, Theorem 94.17.3. The assumptions of Groupoids in Spaces, Lemma 78.25.3 are all satisfied and it follows that we have a $2$-fibre square

It follows from this and Lemma 100.3.1 that $[Z/R_ Z] \to [U/R]$ is representable by algebraic spaces, whereupon it follows from Lemma 100.3.3 that the right vertical arrow is an immersion (resp. closed immersion, resp. open immersion) if and only if the left vertical arrow is. $\square$