How to write the covariant power set functor $$P : \mathsf{FinSet} \to \mathsf{Set}$$ concisely as a colimt of representable functors? There is an epimorphism $$\coprod_{n \geq 0} \hom(\{1,\dotsc,n\},-) \to P,$$ mapping $f : \{1,\dotsc,n\} \to X$ to $\mathrm{im}(f) \in P(X)$. This already provides a generating set of $P$. Compare this with the contravariant power set functor, which is already a representable functor.

This question is motivated by the exercise to find all morphisms of functors $P \to P$ without too much calculations.

How to write the covariant power set functor (restricted to finite sets for simplicity) $$P : \mathsf{FinSet} \to \mathsf{Set}$$ concisely as a colimt of representable functors? There is an epimorphism $$\coprod_{n \geq 0} \hom(\{1,\dotsc,n\},-) \to P,$$ mapping $f \in \hom(\{1,\dotsc,n\},X)$ to $\mathrm{im}(f) \in P(X)$. This already provides a generating set of $P$. Compare this with the contravariant power set functor, which is already a representable functor.

This question is motivated by the exercise to find all morphisms of functors $P \to P$ without too much calculations.