Let $\mathbb D$ be a sound limit doctrine. 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 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.)

Let $\mathbb D$ be a sound limit doctrine. 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 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? (In particular that hold for the doctrines of $\lambda$-small limits and of finite products.)

Let $\mathscr C$ be a locally strongly presentable category (i.e. a cocomplete category that is the free cocompletion under sifted colimits of some small category), and let $X \in \mathscr C$. Is it the case that the coslice $X/\mathscr C$ is also locally strongly presentable?

The analogous property holds for local $\kappa$-presentability (Proposition 1.57 of Adámek–Rosicky's Locally presentable and accessible categories), but does not generally hold for the general notion of $\mathbb D$-accessibility for a sound limit doctrine $\mathbb D$ (Remark 83 of Centazzo's Generalised algebraic models). The proof in AR for traditional accessibility makes use of syntactic arguments (for instance Example 1.41), which do not have obvious analogues for other notions of accessibility, so the conceptual reason the coslice result holds for $\kappa$-presentability is unclear to me.

Let $\mathbb D$ be a sound limit doctrine. 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 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.)

Let $\kappa$ be a regular cardinal, let $\mathscr C$ be a locally strongly $\kappa$-presentable category (i.e. a cocomplete category that is the free cocompletion under $\kappa$-sifted colimits of some small category), and let $X \in \mathscr C$. Is it the case that the coslice $X/\mathscr C$ is also locally strongly $\kappa$-presentable?

The analogous property holds for local $\kappa$-presentability (Proposition 1.57 of Adámek–Rosicky's Locally presentable and accessible categories), but does not generally hold for the general notion of $\mathbb D$-accessibility for a sound limit doctrine $\mathbb D$ (Remark 83 of Centazzo's Generalised algebraic models). The proof in AR for traditional accessibility makes use of syntactic arguments (for instance Example 1.41), which do not have obvious analogues for other notions of accessibility, so the conceptual reason the coslice result holds for $\kappa$-presentability is unclear to me.

Let $\mathscr C$ be a locally strongly presentable category (i.e. a cocomplete category that is the free cocompletion under sifted colimits of some small category), and let $X \in \mathscr C$. Is it the case that the coslice $X/\mathscr C$ is also locally strongly presentable?

The analogous property holds for local $\kappa$-presentability (Proposition 1.57 of Adámek–Rosicky's Locally presentable and accessible categories), but does not generally hold for the general notion of $\mathbb D$-accessibility for a sound limit doctrine $\mathbb D$ (Remark 83 of Centazzo's Generalised algebraic models). The proof in AR for traditional accessibility makes use of syntactic arguments (for instance Example 1.41), which do not have obvious analogues for other notions of accessibility, so the conceptual reason the coslice result holds for $\kappa$-presentability is unclear to me.