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Nov 7, 2015 at 7:34 comment added David Carchedi The particular case you care about is covered by the fact that $Shv(C/c)\simeq Shv(C)/y(c),$ where $y$ is the Yoneda embedding. With this insight, the geometric morphism you seek IS in HTT; it's an etale geometric morphism. The fact I claimed is Proposition 2.2.1 here: arxiv.org/abs/1312.2204
Oct 30, 2015 at 8:49 vote accept Daniel Grady
Oct 29, 2015 at 14:23 comment added Bas Spitters It might be valuable to record something in the nlab
Oct 28, 2015 at 10:47 answer added AAK timeline score: 14
Oct 28, 2015 at 8:27 history asked Daniel Grady CC BY-SA 3.0