Warning 9.2.9.6. In the situation of Theorem 9.2.9.4, the assumption $\kappa \trianglelefteq \lambda $ cannot be omitted. For example, suppose that $\mu $ satisfies $\kappa \trianglelefteq \mu $ and $\lambda \trianglelefteq \mu $ (these condition are satisfied, for example, if $\mu = (2^{\lambda })^{+}$: see Corollary 9.1.7.9). If $S$ is a set of cardinality $\geq \mu $, then Example 9.2.3.11 supplies equivalences
In this case, the conclusion of Theorem 9.2.9.4 is satisfied if and only if the comparison map $\iota : \operatorname{Ind}_{\kappa }^{\lambda }( P_{< \kappa }(S) ) \hookrightarrow P_{< \lambda }(S)$ induces an equivalence of $\operatorname{Ind}_{\lambda }^{\mu }$-completions. If this condition is satisfied, then $\iota $ itself is an equivalence (see Exercise 9.2.6.14), so that $\kappa \trianglelefteq \lambda $ (Example 9.2.3.11).