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Given such that is 0-truncated, then is strongly -connected over .
We have an essential geometric morphism given by a composite of adjoint triples
where the top pairs come from the formula (here) for localization of adjunctions to slices, and the bottom one exists in each case by the adjoint (∞,1)-functor theorem, since the middle one preserves (∞,1)-colimits (since colimits in slices are computed on the dependent sums, since preserves colimits, and since pullbacks preserve colimits in an -topos). The fact that two top composite preserves the terminal object follows now by the idempotency of the various adjunctions and then by infinitesimal cohesion . Finally using that is 0-connected, hence a set it follows from that the composite right adjoint is fully faithful over over , hence is fully faithful on all of .
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