The Clowder Project

where:

• Action on Objects. For each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
  \[ \webleft [\webleft (-\webright )^{+}\webright ]\webleft (X\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X^{+}, \]

  where $X^{+}$ is the pointed set of Definition 4.4.1.1.1.

• Action on Morphisms. For each morphism $f\colon X\to Y$ of $\mathsf{Sets}$, the image
  \[ f^{+}\colon X^{+}\to Y^{+} \]

  of $f$ by $\webleft (-\webright )^{+}$ is the map of pointed sets defined by

  \[ f^{+}\webleft (x\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} f\webleft (x\webright ) & \text{if $x\in X$,}\\ \star _{Y} & \text{if $x=\star _{X}$.} \end{cases} \]