Warning 4.7.4.10. The implications $(1) \Rightarrow (2) \Leftrightarrow (3)$ of Proposition 4.7.4.9 are valid for an arbitrary infinite cardinal $\kappa $. However, the implication $(2) \Rightarrow (1)$ is false if $\kappa $ has countable cofinality (for example, if $\kappa = \aleph _0$).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$