If $\mathcal C$ is a $\kappa$-accessible 1-category, then the category of morphisms $Mor \mathcal C$ is a $\kappa$-accessible 1-category, with the $\kappa$-presentable objects being those morphisms whose domains and codomains are each $\kappa$-presentable.
In the context of $\infty$-categories, the best result I know of is HTT Proposition 220.127.116.11, which shows that if $\mathcal C$ is a $\kappa$-accessible $\infty$-category and $\kappa \ll \tau$ (meaning that $\lambda < \tau \Rightarrow \kappa^\lambda < \tau$ and $\kappa < \tau$), then $Mor \mathcal C$ is $\tau$-accessible.
Lurie's proof, via HTT Lemma 18.104.22.168 (note that this lemma's proof has been revised since the printed edition), seems to really use the full strength of the assumption $\kappa \ll \tau$. Can this be improved to $\kappa = \tau$? Or at least to the "sharply below" relation $\kappa \triangleleft \tau$ familiar from the theory of accessible 1-categories?
This boils down to asking: if $\mathcal C$ is $\kappa$-accessible, then is every morphism of $\mathcal C$ a levelwise $\kappa$-filtered colimit of morphisms between $\kappa$-presentable objects?
In the case of 1-categories, a follow-your-nose argument works: you just take colimiting diagrams for the domains and codomains and factor the original map through stages of the colimit. I suspect that the same must be true in $\infty$-categories, with the same argument in principle working. But the question seems to be much more subtle $\infty$-categorically.