If $\lambda, \mu$ are regular cardinals, say that $\lambda \trianglelefteq \mu$ if $\lambda \leq \mu$ and $$\forall X, \, |X| < \mu \implies \mathrm{cf} (P_\lambda(X)) < \mu$$ Here $P_\lambda(X)$ is the poset of subsets of $X$ of cardinality $<\lambda$, and $\mathrm{cf}$ takes the cofinality of a poset.

Then $\trianglelefteq$ is an $\mathrm{Ord}$-directed partial order on regular cardinals which is strictly weaker than the ordering by size (by $\mathrm{Ord}$-directed, I mean that any set of cardinals has a $\triangleleft$-upper bound). For instance, $\aleph_1 \not \trianglelefteq \aleph_{\omega+1}$ but $\aleph_0 \trianglelefteq \mu$ for all $\mu \geq \aleph_0$.

This relation comes up in the theory of accessible categories, because the following are equivalent when $\lambda \leq \mu$ are regular cardinals:

- Every $\lambda$-accessible category is $\mu$-accessible.
- The category $\mathrm{Pos}_\lambda$ of $\lambda$-directed posets and embeddings [which is always $\lambda$-accessible] is $\mu$-accessible.
- For every $\lambda$-directed poset $P$, the $\lambda$-directed subsets are cofinal in $P_\mu(P)$.
- $\lambda \trianglelefteq \mu$.

A good reference is Adámek and Rosický, though these facts mostly go back at least to Makkai and Paré. The only facts I know how to use to determine whether $\lambda \triangleleft \mu$ are the following:

- $\lambda \triangleleft \lambda^+$
- If $\left(\alpha <\lambda, \beta < \mu \implies \beta^\alpha < \mu\right)$, then $\lambda \triangleleft \mu$ for the somewhat boring reason that for $|X| < \mu$ we have $|P_\lambda(X)| < \mu$ so that certainly $\mathrm{cf} (P_\lambda(X)) < \mu$.
- If $\lambda < \mu$ and $\mathrm{cf} \mu \leq \lambda$, then $\lambda^+ \not \triangleleft \mu^+$.

**Questions**
I basically would just like to see what the set theorists (and category theorists) around here might have to say about the relation $\triangleleft$. Is this by chance a well-studied relation? Should I expect the question of whether $\lambda \triangleleft \mu$ holds to be mostly determined, or mostly independent, given reasonable descriptions of $\lambda$ and $\mu$? Should I expect (1), (2), and (3) to pretty much determine what can be determined, or should I expect there's more to say? I suppose things are not too complicated under GCH, for example. Does having good control over the $\beth$ function in general give good control over the functions $\mathrm{cf}(P_\lambda(\beta))$?

A really basic question: is the quantity $\mathrm{cf}(P_\lambda(X))$ (weakly) monotonic in $|X|$?