Counterpoint: The covariant powerset functor is a monad in a fairly natural way (the monad's multiplication $\mathcal{P}^2(X)\to\mathcal{P}(X)$ takes a set of subsets of $X$ to their union), and the Kleisli category for this monad is the category of sets-with-relations-as-morphisms (=:“swram”s). Which means we can see the powerset functor as the composition of two adjoints: the forgetful functor from swrams, and its right adjoint (which on objects takes a swram to the powerset). I think this makes the covariant powerset functor as “natural” as the category of swrams, which I find appealing.
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