In the standard reference books *Locally presentable and accessible categories* (Adamek-Rosicky, Theorem 2.11) and *Accessible categories* (Makkai-Pare, $\S$2.3), it is shown that for regular cardinals $\lambda\le\mu$, the following are equivalent:

- Every $\lambda$-accessible category is $\mu$-accessible.
- For every $\mu'<\mu$, the set $P_\lambda(\mu')$ of subsets of $\mu'$ of cardinality $<\lambda$ has a cofinal subset of cardinality $<\mu$.

This relation is denoted $\lambda\unlhd \mu$ (or $\lambda \lhd \mu$ for the irreflexive version).

In *Higher Topos Theory* (Lurie, Definition A.2.6.3) the relation $\lambda\ll\mu$ is defined to mean

- For every $\lambda'<\lambda$ and $\mu'<\mu$ we have $(\mu')^{\lambda'} <\mu$.

By Example 2.13(4) of Adamek-Rosicky, if $\lambda\ll\mu$ and $\lambda<\mu$ then $\lambda\lhd \mu$. (Indeed, in this case $P_\lambda(\mu')$ itself has cardinality $<\mu$.)

Does the converse hold? That is, if $\lambda\lhd \mu$ do we have $\lambda\ll\mu$?

Note that the two relations definitely differ in the reflexive case: $\lambda\unlhd\lambda$ is always true, but $\lambda\ll\lambda$ holds only when $\lambda$ is inaccessible. It also seems that the converse implies the generalized continuum hypothesis for regular cardinals, since if $\lambda^+ < 2^\lambda$ then we have $\lambda^+ \lhd \lambda^{++}$ but not $\lambda^+ \ll \lambda^{++}$. Thus, the converse is not provable in ZFC. Is it disprovable?