Let $\kappa$ be a small regular cardinal and $D$ a locally $\kappa$-presentable category. Is it true that $D$ is also locally $\tau$-presentable for any $\tau > \kappa.$ Adamek und Rosicky show in "Locally Presentable and accessible categories" that this is not true for general accessible categories (but holds if $\tau >>\kappa$). So I ask the question for locally presentable categories.
I have the following proof in mind but wonder if there is a gap: Let $D$ be a locally $\kappa$-presentable category. Then $D$ is $\mathrm{Ind}_\kappa(D^\kappa)$, where $D^\kappa$ is the full subcategory of $D$ spanned by the $\kappa$-compact objects. Because $D$ admits small colimits, $D^\kappa$ (which is closed in D under $\kappa$-small colimits) admits $\kappa$-small colimits. This implies that $\mathrm{Ind}_\kappa(D^\kappa)$ is a reflexive full subcategory of $P(D^\kappa)$ closed under $\kappa$-filtered colimits, where $P(D^\kappa)$ is the category of presheaves on $D^\kappa$.
It is well known that for every small regular cardinal $\tau$ every reflexive full subcategory closed under $\tau$-filtered colimits of a locally $\tau$-presentable category is again locally $\tau$-presentable. It is also well-known that the category of presheaves on any small category is locally $\tau$-presentable for any small regular cardinal $\tau:$ every $\tau$-small colimit of representables is $\tau$-compact and every presheaf is a $\tau$-filtered colimit of $\tau$-small colimits of representables. Because $\mathrm{Ind}_\kappa(D^\kappa)$ is closed under $\kappa$-filtered colimits in $P(D^\kappa)$, it is also closed under $\tau$-filtered colimits in $P(D^\kappa)$ for any $\tau> \kappa.$ So the category $\mathrm{Ind}_\kappa(D^\kappa)$ is locally $\tau$-presentable.
