This might be a naive question, but it came up from a discussion on social media regarding duality. It would be nice to have a somewhat elementary interpretation of what the dual space of a vector space/module is.
A naive attempt was to consider the universal property (I have no idea how to make commutative diagrams here, so bear with me please) that on any index set $I$, with a ring $R$, there is the free module $F_R(I)$ given by taking formal $R-$linear combinations of elements of $I$. Now we have a canonical inclusion $j: I \hookrightarrow F_R(I)$, and if we have any map to any other $R$ module $M$, $\phi: I \to M$, then there is a unique map taking $F_R(I) \to M$ by just sending the canonical basis for $F_R(I)$ to wherever they are sent to in $M$, and checking this is a well-defined morphism.
Now, this means there is a category where the objects are maps $I \to M$, $M$ now an arbitrary $R$-module, and the morphisms are commutative triangles. My initial hope was that since the free module above is initial in this category, that the dual might be the dual space. This seems to be false. There is a seemingly also canonical map (it might not be, I guess) map $I \to \operatorname{Hom}_R(F_R(I),R)$ taking the elements of $I$ to the relative 'functionals', the guys which take value $1$ on their image in the above inclusion and are $0$ otherwise. So one might hope that this is the terminal object in this category. However it appears not to be, since a generic map $I \to M$ for some $M$ might not be injective, so I don't know how I'd define the map in this case, as under the inclusion into the dual I just wrote down, these would need to go to different places. So this isn't the dual in this category.
A bit more thinking leads me to believe the only way to circumvent this problem is if the terminal object in this category were actually the $0$ module. So not only does this not work, it's not even close.
So now I'm back to square one, except I want to recruit the people more knowledgable of algebra than me on this site. I'm seeking a universal property of dual spaces which should make the dual space the terminal object of some category. It would be ideal if the case of vector spaces was relatively simple, so that I could go back to my discussion with my friends and tell them what I've learned, or link them this thread (and they're largely undergraduates), but if it's not, that's fine too.
Turns out after a whole bunch of years of doing linear algebra, there are elementary questions I still don't understand!