The contravariant endofunctor $$\mathbf{Set}^{op} \rightarrow \mathbf{Set}$$ $$X \mapsto [X,2]$$ can be made into a covariant endofunctor $$\mathbf{Set} \rightarrow \mathbf{Set}$$ $$X \mapsto \mathcal{P}(X)$$ in such a way that: these functors do the same thing to objects they do the same thing to isomorphisms, excepting that $X \mapsto [X,2]$ flips the direction. This isn't too surprising, since $X \mapsto [X,2]$ can be seen as "completing" a set to a suplattice, and then forgetting the suplattice structure. It's probably fair to say that the covariant structure on this functor comes essentially comes from this fact (though perhaps there is a better way at looking at it). Anyway, I'm wondering if we can do something similar to the dual-vectorspace functor $$\mathbb{R}\mathbf{Mod}^{op} \rightarrow \mathbb{R}\mathbf{Mod}$$ $$X \mapsto \mathbb{R}\mathbf{Mod}(X,\mathbb{R}).$$ It seems likely that we can, since I tend to think of dual vector spaces as "completing" a vector space to a gadget in which certain infinite sums exist, and then forgetting this extra structure. Login To add answer/comment