Remark 3.3.4.4 (Functoriality). The morphisms $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$ and $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ depend functorially on the simplicial set $X$. That is, for every map of simplicial sets $f: X \rightarrow Y$, the diagrams
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{ \rho _ X } \ar [d]^{f} & \operatorname{Ex}(X) \ar [d]^{ \operatorname{Ex}(f) } & \operatorname{Sd}(X) \ar [r]^-{ \lambda _ X} \ar [d]^{ \operatorname{Sd}(f) } & X \ar [d]^{f} \\ Y \ar [r]^-{ \rho _ Y } & \operatorname{Ex}(Y) & \operatorname{Sd}(Y) \ar [r]^-{ \lambda _ Y } & Y } \]
are commutative. We may therefore regard the constructions $X \mapsto \rho _{X}$ and $X \mapsto \lambda _{X}$ as natural transformations of functors
\[ \rho : \operatorname{id}_{ \operatorname{Set_{\Delta }}} \rightarrow \operatorname{Ex}\quad \quad \lambda : \operatorname{Sd}\rightarrow \operatorname{id}_{ \operatorname{Set_{\Delta }}}. \]