Let $p : E \to X$ be an (∞,1)-functor of (∞,1)-categories. A cartesian section of $p$ is a section $\sigma : X \to E$ that sends all 1-morphisms in $X$ to Cartesian morphisms in $E$.
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If $p : E \to X$ is a Cartesian fibration classified by an (∞,1)-functor $F : X \to (\infty,1)Cat^{op}$ then $\Gamma_X^{cart}(E)$ is equivalent to the limit of $F$
See the discussion at limit in a quasi-category for details.
In corollary 3.3.3.2 of
the collection of cartesian sections of $p : E \to X$ appears as $Maps_X^\flat(X^#, E^{cart})$.
Here
the simplicial set $Maps_X^\flat(\cdots)$ is the simplicial set underlying the internal hom of marked simplicial sets over $X$ (beginning of section 3.1.3);
$X^#$ is the simplicial set $X$ with all cells marked (beginning of section 3.1)
and $E^{cart}$ is $E$ with precisely all Cartesian morphisms marked (def. 3.1.1.9).