ct.category theory – Co-Cartesian vs. Coherent Fibrations local coartesians

Tell $$pi: C a J$$ it is an internal fibration of $$infty$$-categories Then "morally", $$pi$$ corresponds to a diagram indexed by $$J$$ in the "category of categories with correspondences", and if $$pi$$ is cocartesian then corresponds to a diagram indexed by $$J$$ in the "category of categories with functors".

From this, it would be natural to guess that the CoCartesian condition is the requirement that the correspondence on each arrow in $$J$$ It is representable by a functor. This is almost but not quite right. These fibrations are denominated "locally coCartesianas" by Lurie and in chapter 2.4.2 of Higher Topos Theory, offers several ways to quantify the "almost" in question, that is to say, the conditions of a local coCartesian fibration that do it coCartesiano.

A condition that I could not find in HTT (maybe because it's obvious) but that seems enough is the following:

Guess. An internal fiber $$C a J$$ is coCartesian if the fibrations $$C ([n]) a J ([n]$$ They are locally co-Cartesians for all. $$n$$, where $$C ([n]$$ the category of functors of the $$n$$-simply to $$C$$.

In fact, it seems that it should be enough to impose this condition only for $$n = 0, 1$$.

My question: is this conjecture correct or solvable? And, can a similar criterion be given for a locally sustainable arrow? $$f$$ of $$J$$ to be a co-supporter of society (maybe something that involves the regression of $$C$$ to the over category of $$f$$ in $$J$$)?