Remark 3.3.3.7. The homeomorphisms $f_{X}: | \operatorname{Sd}(X) | \simeq |X|$ constructed in the proof of Proposition 3.3.3.6 are characterized by the following properties:
In the special case where $X = \Delta ^ n$ is a standard simplex, $f_{X}$ is given by the composition
\[ | \operatorname{Sd}( \Delta ^ n) | \simeq | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \xrightarrow {f} | \Delta ^ n |, \]where the first map is supplied by the identification $\operatorname{Sd}( \Delta ^{n} ) \simeq \operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ of Example 3.3.3.5 and $f$ is the homeomorphism of Proposition 3.3.2.3.
Let $u: X \rightarrow Y$ be a morphism of braced simplicial sets which carries nondegenerate simplices of $X$ to nondegenerate simplices of $Y$. Then the diagram of topological spaces
\[ \xymatrix@R =50pt@C=50pt{ | \operatorname{Sd}(X) | \ar [r]^-{ f_ X }_-{\sim } \ar [d]^{ | \operatorname{Sd}(u) |} & | X | \ar [d]^{ | u | } \\ | \operatorname{Sd}(Y) | \ar [r]^-{ f_ Y}_-{\sim } & | Y | } \]commutes.