Remark 3.3.2.4 (Functoriality). Let $\alpha : [m] \rightarrow [n]$ be a nondecreasing function between partially ordered sets, so that $\alpha $ induces a nondecreasing map $\operatorname{Chain}[\alpha ]: \operatorname{Chain}[m] \rightarrow \operatorname{Chain}[n]$ (Remark 3.3.2.2). If $\alpha $ is injective, then the diagram of topological spaces
\[ \xymatrix@R =50pt@C=50pt{ | \operatorname{N}_{\bullet }( \operatorname{Chain}[m] ) | \ar [d]^{ | \operatorname{N}_{\bullet }( \operatorname{Chain}[\alpha ] ) |} \ar [r]^-{ f_ m }_-{\sim } & | \Delta ^{m} | \ar [d]^{ | \alpha | } \\ | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \ar [r]^-{ f_ n }_-{\sim } & | \Delta ^{n} | } \]
is commutative, where the horizontal maps are the homeomorphisms supplied by Proposition 3.3.2.3. Beware that if $\alpha $ is not injective, this this diagram does not necessarily commute. For example, the induced map $| \Delta ^{m} | \rightarrow | \Delta ^{n} |$ carries the barycenter of $| \Delta ^{m} |$ to the point
\[ ( \frac{ | \alpha ^{-1}(0) |}{m+1}, \frac{| \alpha ^{-1}(1) |}{m+1}, \ldots , \frac{| \alpha ^{-1}(n) |}{m+1}) \in | \Delta ^{n} |, \]
which need not be the barycenter of any face $| \Delta ^{n} |$.