Remark 3.3.2.2 (Functoriality). Let $f: Q \rightarrow Q'$ be a nondecreasing map between partially ordered sets. Then $f$ induces a map $\operatorname{Chain}[f]: \operatorname{Chain}[Q] \rightarrow \operatorname{Chain}[Q']$, which carries each nonempty linearly ordered subset $S \subseteq Q$ to its image $f(S) \subseteq Q'$. By means of this construction, we can regard $Q \mapsto \operatorname{Chain}[Q]$ as functor from the category of partially ordered sets to itself.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$