The Clowder Project

where:

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

  where $X^{-}$ is the set of Definition 4.4.2.1.1.

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

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

  \[ f^{-}\webleft (x\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (x\webright ) \]

  for each $x\in X^{-}$.