Let S be a Grothendieck site, the (either left or right adjoint to the) Grothendieck construction assigns to each groupoid fibration over S a presheaf valued in groupoids. The following feels it might be morally correct:

A prestack is a stack $\iff$ its corresponding presheaf of groupoids is a sheaf

Is this true? Does the Grothendieck construction map descent data for stacks into descent data for sheaves?

(Note I haven’t put too much thought into this yet so the question might be silly or trivial)