$\infinity$-Stackification is another term for (∞,1)-sheafification. It is the direct (∞,1)-categorical analog of the following 1-categorical situation.

which sends every presheaf$F$ on $S$ to another presheaf$\bar F$ which is weakly equivalent to $F$ with respect to the homotopical category structure on $PSh(S)$ induced from the Grothendieck topology on $X$. The presheaf $\bar F$ respects weak equivalences and satisfies descent in that the hom-functor$Hom_{PSh(S)}(-,\bar F)$ sends weak equivalences (the local isomorphisms) to weak equivalences.