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