I am trying to formulate an example that the contravariant powerset functor is not full.
Define the contravariant powerset functor $\mathcal{P}:{\text{Set}} \to {\text{Set}}$ as the functor which sends a morphism $f : X \to Y$ to the inverse mapping $\mathcal{P}f: \mathcal{P}Y \to \mathcal{P}X$, where for $B \subseteq Y$, $\mathcal{P}f$ is the preimage $f^{-1}[B]=\{x \in X \mid f(x) \in B\}$.
For a counter example, can I consider the map $u: \mathcal{P} \emptyset \to \mathcal{P}\{0,1\}$, defined by $u(\emptyset) = B$, for $B \in \mathcal{P}\{0,1\}$, and say there exists no function from $ \{0,1\} $ to $\emptyset$, hence the contravariant powerset functor is not full?