Definition 9.1.7.5 (Makkai-Paré [MR1031717]). Let $\kappa $ and $\lambda $ be regular cardinals. We write $\kappa \trianglelefteq \lambda $ if $\kappa \leq \lambda $ and the following additional condition is satisfied:
For every $\lambda $-small set $S$, there exists a $\lambda $-small collection of subsets $\{ S_ i \subseteq S \} _{i \in I}$ such that each $S_ i$ is $\kappa $-small, and every $\kappa $-small subset of $S$ is contained in some $S_ i$.
We write $\kappa \triangleleft \lambda $ if $\kappa \triangleleft \lambda $ and $\kappa \neq \lambda $.