Power set functors

There are two functors vying for the name "power set functor".

For the first, define $\mathcal{P}:\mathbf{S}\mathbf{e}\mathbf{t}\to \mathbf{S}\mathbf{e}\mathbf{t}$ as follows:

• On objects: for each set $X$, define $\mathcal{P}(X)$ to be the power set of $X$, i.e., the set of all subsets of $X$.
• On arrows: for each set map $f:X\to Y$, define a set map $\mathcal{P}(f):\mathcal{P}(X)\to \mathcal{P}(Y)$ by sending each subset $S\subseteq X$ to its image $f(S)\subseteq Y$.

For the second, define ${\mathcal{P}}^{\prime }:{\mathbf{S}\mathbf{e}\mathbf{t}}^{\text{op}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ as follows^[1]:

• On objects: for each set $X$, define ${\mathcal{P}}^{\prime }(X)$ to once again be the power set of $X$.
• On arrows: for each arrow $f^{\text{op}}:X\to Y$ (corresponding to a set map $f:Y\to X$), define a set map ${\mathcal{P}}^{\prime }(f^{\text{op}}):{\mathcal{P}}^{\prime }(X)\to {\mathcal{P}}^{\prime }(Y)$ by sending each subset $S\subseteq X$ to its preimage $f^{-1}(X)\subset Y$. (Recall that $f^{-1}(X)=\{y\in Y\mid f(y)\in X\}$.)

Verify that $\mathcal{P}$ and ${\mathcal{P}}^{\prime }$ are indeed both functors. The first is sometimes called the covariant power set functor and the second the contravariant power set functor.