@TimCampion This is discussed in Rezk's notes on quasicategories, section 31

@TimCampion this is only for intuition and probably very circular, but if you write your category as a colimit of points and arrows, then you only need to know that projection from a limit to the limit over the vertices is conservative in Cat and to prove this for Delta^0 and Delta^1.

And of course, I assume here also that Fun(-,C) takes colimits to limits...

@TimCampion an arrow is an equivalence iff it is a colimit diagram when we think of [1]=[0]^{\triangleright}. So this is a special case of the fact that colimits are detected pointwise (HTT.5.1.2.2). Admittedly that’s a bit awkward... another tactic would be to reduce to the case of presheaves then straighten to right fibrations and then use that equivalences of right fibrations are detected fiberwise.