An ∞-stack on a (∞,1)-category-domain C that happens to be an ordinary category (i.e. not a derived stack) is rectified if it is an ordinary functor C op→ SSet instead of a general (∞,1)-functor (i.e. pseudofunctor).
A central theorem about the model structure on simplicial presheaves says that rectified ∞-stacks are sufficient: they already present the full (∞,1)-category of (∞,1)-sheaves (= ∞-stacks).
Notice that, by a result recalled at descent for simplicial presheaves, a rectified ∞-stack A is an ∞-groupoid internal to (pre)sheaves satisfying a descent condition.