冪集合函手

$ \mathrm{Pow}_\circ\colon \mathbf{Set}→\mathbf{Set}

$ X.\mathrm{Pow}_\circ = 2^X

$ 2^X = \left\{ X' \mid X' \subseteq X \right\}

$ F_\triangleright \coloneqq F.\mathrm{Pow}_\circ \colon 2^X→2^Y

$ X'.F_\triangleright = \left\{ x'.F \in Y \mid x' \in X' \right\}

$ =\left\{ y \in Y \mid \exist x' \in X,\ y = x'.F \right\}

where

$ X,Y \colon \mathbf{Set}

$ F\colon X→Y

$ X' \colon 2^X

$ \mathrm{Pow}_\circ\colon \mathbf{Set}→\mathbf{B.Lattice}

$ \mathrm{Pow}^\circ\colon \mathbf{Set}^{\mathrm{op}}→\mathbf{Set}

$ X.\mathrm{Pow}^\circ = 2^X

$ F^\triangleleft \coloneqq F.\mathrm{Pow}^\circ \colon 2^Y→2^X

$ Y'.F^\triangleleft = \left\{ x \in X \mid \ x.F \in Y' \right\}

$ = \left\{ x \in X \mid \ \exists y' \in Y', y' = x.F \right\}

where

$ X,Y \colon \mathbf{Set}

$ F\colon X→Y

$ Y' \colon 2^Y

$ \mathrm{Pow}^\circ\colon \mathbf{Set}^{\mathrm{op}}→\mathbf{B.Lattice}