Let $\kappa$ be a regular cardinal. A category $\mathscr C$ is locally $\kappa$-presentable iff it is the free completion of a small $\kappa$-cocomplete category under $\kappa$-filtered colimits. Is there a known characterisation of the categories $\mathscr C$ that are:

- locally $\kappa$-presentable and cartesian-closed;
- locally $\kappa$-presentable and locally cartesian-closed;

in terms of being the free cocompletion of a small $\kappa$-cocomplete category *with particular structure* under $\kappa$-filtered colimits?