The first condition says that it is an (∞,1)-sheaf with values in 2-groupoids. The second says that $P \to *$ is an effective epimorphism and that the 0-th homotopy sheaf is the terminal sheaf. In the literature this is often stated as saying that $P$ is a) locally connected and b) locally non-empty .

Abelian 2-gerbes

For $\mathcal{X}$ an (∞,1)-topos, an abelian 2-gerbe$P$ in $\mathcal{X}$ is an object which is