In an ordinary category $C$, one says that an object $X$ is $\kappa$-compact if the representable functor $Hom(X,-)\colon C \to Set$ preserves $\kappa$-filtered colimits. We say $C$ is locally presentable if it is cocomplete and "generated" by $\kappa$-compact objects for some $\kappa$.

In an $(\infty,1)$-category $C$, one says that an object $X$ is $\kappa$-compact if the representable functor $Hom(X,-)\colon C \to \infty Gpd$ preserves $\kappa$-filtered $(\infty,1)$-colimits. We say $C$ is locally presentable if it is cocomplete and "generated" by $\kappa$-compact objects for some $\kappa$.

There are many equivalent, also analogous, definitions in both cases.

An $(\infty,1)$-category is locally presentable if and only if it admits a presentation by some locally presentable, cofibrantly generated model category. However, the only proof of this fact that I have seen (in A.3.7.6 in *Higher Topos Theory*) uses a different equivalent definition of both notions (as an accessible localization of some presheaf category). Thus my question:

Is there any relationship between an object $X$ being $\kappa$-compact in a locally presentable model category and being $\kappa$-compact in the $(\infty,1)$-category that it presents?