Functoriality. The assignments $A,\webleft (X,x_{0}\webright ),\webleft (A,\webleft (X,x_{0}\webright )\webright )$ define functors
\[ \begin{array}{ccc} A\pitchfork -\colon \mkern -15mu & \mathsf{Sets}\mathrlap {{}_{*}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -\pitchfork X\colon \mkern -15mu & \mathsf{Sets}^{\mathrlap {\mathsf{op}}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -_{1}\pitchfork -_{2}\colon \mkern -15mu & \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}. \end{array} \]
In particular, given:
- A map of sets $f\colon A\to B$;
- A pointed map $\phi \colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$;
the induced map
\[ f\odot \phi \colon A\pitchfork X\to B\pitchfork Y \]
is given by
\[ \webleft [f\odot \phi \webright ]\webleft (\webleft [\webleft (x_{a}\webright )_{a\in A}\webright ]\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\phi \webleft (x_{f\webleft (a\webright )}\webright )\webright )_{a\in A}\webright ] \]
for each $\webleft [\webleft (x_{a}\webright )_{a\in A}\webright ]\in A\pitchfork X$.