Powerset functor

A powerset functor, 𝒫 : 𝗦𝗲𝘁 → 𝗦𝗲𝘁, sends subsets to their image under maps.

∀A,B ∈ Obj(𝗦𝗲𝘁), f : A → B and S ⊂ A: 𝒫 A = 𝒫(A), 𝒫 f : 𝒫(A) → 𝒫(B), S ⟼ f(S)

𝗦𝗲𝘁 is a category of sets. The objects are all small sets, and the arrows are functions between sets.