Rank of presentability of internal Hom of locally presentable categories

Question

Let $C$ and $D$ be locally $\kappa$-presentable categories. It is written on the nLab that the category $\mathrm{Ladj}(C, D)$ of cocontinuous functors from $C$ to $D$ is again locally $\kappa$-presentable. Is this actually true? The proof of the local presentability of $\mathrm{Ladj}(C, D)$ that I know (the one used to prove Corollary 3.18 in Bird's PhD thesis) yields a rank of presentability that depends on more than just the ranks of presentability of $C$ and $D$.

However, I also was not able to come up with a counterexample to the claim. The simplest case in which things could go wrong is when $C$ is the category of presheaves on a category $A$ of cardinality at least $\kappa$. But in this case, $\mathrm{Ladj}(C, D) = [A, D]$ actually is $\kappa$-presentable. So $C$ will need to be a reflective subcategory of a presheaf category, but I was unable to construct such an example where I could compute that the rank of presentability of $\mathrm{Ladj}(C, D)$ exceeds $\kappa$.

Answer 1

Let $C$ be a locally $\kappa$-presentable category. Then we may write $C = |C_\bullet|$ where each $C_n$ is a presheaf category and the geometric realization is taken in $Pr^L_\kappa$ (and so also is a geometric realization in $Pr^L$). So if $D$ is $\kappa$-presentable, then $LAdj(C,D) = LAdj(|C_\bullet|,D) = Tot LAdj(C_\bullet,D)$, where the totalization is taken in $Pr^L$ (and the cosimplicial diagram lives in $Pr^L_\kappa$ [1]). Now, $Pr^L_\kappa$ is closed in $Pr^L$ under $\kappa$-small limits if $\kappa$ is uncountable, and the simplex category is countable. So if $\kappa$ is uncountable, then $LAdj(C,D)$ is again $\kappa$-presentable. ( 1-categorically, I suppose this might even work if $\kappa = \omega$, since the cosimplicial object can be truncated at a finite stage -- but I'm not sure that $Pr^L_\omega$ is closed under finite limits in $Pr^L$.)

[1] This is actually the subtlest step, I think. By the presheaf case, $LAdj(C_n,D)$ is locally $\kappa$-presentable. The subtle part is verifying that the transition maps $LAdj(C_n,D) \to LAdj(C_m,D)$ preserve $\kappa$-presentable objects. Note that if $A$ is a small category and $D$ is locally $\kappa$-presentable, then the $\kappa$-presentable objects of $D^A$ are those functors $A \to D$ which are left Kan extended from a functor $B \to D_\kappa$ where $B$ is $\kappa$-small and $D_\kappa \subseteq D$ comprises the $\kappa$-presentable objects. The relevant functors $C_{n+1} \to C_n$ with which we are precomposing all have fully faithful right adjoints $\iota$ [2], so if $C_n \to D$ is extended from $B$, then the composite $C_{n+1} \to C_n \to D$ is extended from $\iota(B)$; thus the precomposition functor $LAdj(C_n,D) \to LAdj(C_{n+1},D)$ does indeed preserve $\kappa$-presentable objects.

[2] I suppose to be sure of this, I should say which simplicial object $C_\bullet$ I'm using. What I have in mind is the simplicial object $C_n = Ind_\kappa (F^n C_\kappa)$, where $C_\kappa \subseteq C$ is the $\kappa$-presentable objects, and $F$ is the monad which freely adjoins $\kappa$-small colimits to a category. Note that $F : Cat \to Cat$ is a monad, and that $C_\kappa$ is an algebra for $F$. Therefore, in $Cat$, the simplicial object $F^\bullet C_\kappa$ admits a split augmentation by $C_\kappa$. Moreover, the monad $F$ is lax-idempotent. Therefore, the degeneracies of the split augmented simplicial object are right adjoint right inverses to the corresponding face maps. Applying the 2-functor $Ind_\kappa$ preserves this property.

Answer 2

$\mathsf{cocont}(A,B)$ is locally presentable but I don't see why it should be locally $\kappa$-presentable.

Let $A$ and $B$ be two locally $\kappa$-presentable categories.

$\mathsf{cocont}(A,B)$ is equivalent to $\kappa\mathsf{-cocont}(\text{Pres}_\kappa(A), B)$. The latter is clearly the category of models of a colimit sketch in a locally presentable category (namely $B$) and thus is locally presentable by Rem. 2.63 in Locally presentable and accessible categories by Adamek and Rosicky.

Following carefully Rem 2.59 there is no way to infer that $\kappa\mathsf{-cocont}(\text{Pres}_\kappa(A), B)$ is $\kappa$-accessible from this argument, even if the cocones have size $\kappa$.