Let $\mathbb D$ be a [sound limit doctrine](https://www.sciencedirect.com/science/article/pii/S0022404902001263). When $\mathbb D$ is the doctrine of $\lambda$-small limits, then for any $\mathbb D$-presentable category $\mathscr C$ and $X \in \mathscr C$, the coslice $X/\mathscr C$ is also $\mathbb D$-presentable (Proposition 1.57 of Adámek–Rosicky's *Locally presentable and accessible categories*). However, this does not hold for general $\mathbb D$ (Remark 83 of Centazzo's [*Generalised algebraic models*](https://pul.uclouvain.be/resources/titles/29303100658430/extras/70957_sciences_centazzov3_1002422.pdf) shows that this property fails for $\mathbb D$ the empty doctrine).

Unfortunately, the proof in AR makes use of syntactic arguments, which do not have obvious analogues for other notions of accessibility, and so it is unclear in what cases this property may or may not hold in general.

Are there any convenient sufficient or necessary conditions on $\mathbb D$ for $\mathscr C$ being $\mathbb D$-presentable to imply $\mathscr C/X$ is $\mathbb D$-presentable? (Ideally that subsume the doctrines of $\lambda$-small limits and of $\lambda$-small products.)