In the paper by Verity and Riehl "Fibrations and Yoneda lemma in an ${\infty}$-cosmos" (https://arxiv.org/abs/1506.05500), they prove a Yoneda lemma that holds in any ${\infty}$-cosmos (see Corollary 6.2.13). Therefore, in particular, it holds in $\textbf{qCat}$.

If we unravel their definition of cartesian fibration, we find out that it is almost representably defined, in the sense that an isofibration $p:E \to B$ is a cartesian fibration in an ${\infty}$-cosmos iff $map(A,E) \to map(A,B)$ is (almost) a cartesian fibration for every $A$. More precisely, instead of the usual requirement of a suitable lift for a certain edge which has to be cartesian with respect to the inner fibration $map(A,p)$,they ask for a weaker property: this lift $\chi$ has to be such that $map(A,p)$ has the right lifting property only against the inclusion $\Lambda^2[2] \to \Delta[2]$ when the restriction to $\Delta^{\{1,2\}}$ is $\chi$ itself (so it is Lurie's definition of cartesian edge where we restrict ourselves to n=2). They also have a conservativity requirement, but I am not exactly sure if and how this fits into my question.

Does this produce an equivalent notion of cartesian fibration (I suspect not)? Or at least, is it true that $p:X \to Y$ inner fibration of quasicategories is a cartesian fibration iff $map(A,p)$ is a cartesian fibration in this weaker sense for every quasicategory (or even any simplicial set) A?