Remark 3.3.4.2. Let $f: P \rightarrow Q$ be a nondecreasing function between partially ordered sets. Then the diagram of partially ordered sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Chain}[P] \ar [r]^-{\mathrm{Max}} \ar [d]^{ S \mapsto f(S) } & P \ar [d]^{f} \\ \operatorname{Chain}[Q] \ar [r]^-{ \mathrm{Max} } & Q } \]
is commutative.