Can somebody help me with a reference showing that the homotopy semicosimplicial totalization of a cosimplicial space is homotopy equivalent to its usual homotopy totalization? Is it because the inclusion of the semisimplicial category into the simplicial category is a homotopy left cofinal functor?

I agree this should be standard, but I've only seen the proof in one place. See Lemma 6.5.3.7 of Lurie's Higher Topos Theory. 

