# When is a locally presentable category (locally) cartesian-closed?

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:

1. locally $$\kappa$$-presentable and cartesian-closed;
2. 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?

I think this is the Day reflection theorem when viewing the LFP as a reflective subcategory of a presheaf topos. For the locally Cartesian closed case, I guess you would just apply some fibred category variant of the Day reflection theorem as each $$\mathcal{E}/X$$ is a reflective subcategory of the Cartesian closed $$[C,Set]/X$$.
• Day's reflection theorem tells us that the LFP category $\mathscr C$ is cartesian-closed iff the reflector is cartesian: how can we characterise the corresponding category of finitely presentable objects for $\mathscr C$ using this? – varkor Oct 24 '20 at 19:25
Prop 4.1. If $$C$$ is an elementary topos, then $$\mathsf{Lex}(C^\circ,\mathsf{Set})$$ is cartesian closed.