Given an infinity topos $\mathcal{E}$, one can pass to its hypercompletion $\widehat{\mathcal{E}}$, which is the full subcategory on those objects local with respect to $\infty$-connected morphisms (morphisms inducing isomorphisms on all homotopy sheaves). Can passing to the hypercompletion be made functorial with respect to all geometric morphisms? (For my purposes, it would suffice to know functoriality with respect to etale ones). Thanks!

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Higher Topos Theory, hypercompletion is right adjoint to the inclusion of hypercomplete $\infty$-topoi into all $\infty$-topoi. Doesn't that make it functorial? – Mike Shulman Mar 1 '13 at 2:48