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Say $\pi: C\to J$ is an inner 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 it 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 over every arrow in $J$ is representable by a functor. This is almost but not quite correct. Such fibrations are called "locally coCartesian" by Lurie and in chapter 2.4.2 of Higher Topos Theory, he gives several ways of quantifying the "almost" in question, i.e. conditions on a locally coCartesian fibration which make it coCartesian.

One condition that I could not find in HTT (maybe because it is obvious) but that seems sufficient is the following:

Conjecture. An inner fibration $C\to J$ is coCartesian if the fibrations $C([n])\to J([n])$ are locally coCartesian for all $n$, where $C([n])$ the category of functors from the $n$-simplex to $C$.

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

My question: is this conjecture correct or fixable? And can one give a similar criterion for a locally coCartesianly liftable arrow $f$ of $J$ to be coCartesianly liftable (maybe something involving the pullback of $C$ to the overcategory of $f$ in $J$)?

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    $\begingroup$ I think your “moral” statement is only true if you ask for flat inner fibrations. $\endgroup$ – Dylan Wilson Jul 12 at 17:58
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    $\begingroup$ (And indeed, a flat inner fibration which is locally cocartesian is cocaresian). $\endgroup$ – Dylan Wilson Jul 12 at 18:02

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