Let $\mathcal{C}$ be a category, and let $A$ and $B$ be objects in that category. It is clear that given a morphism $f:A\to B$, we know what $f$ does on generalized elements $e:A'\to A$, since this is just done by composing with $f$. However, I was wondering if there is always a converse construction. Or, if there are any known constraints on when such a converse construction can be made. That is, can we define a morphism $f:A\to B$, given that we know how generalized elements are mapped to $B$. That is, we know $e:A'\to A$ and we 'know' $f\circ a:A'\to B$.
Thanks!
That's a great question ! The answer is yes, that's essentially the statement of the Yoneda lemma.
It says more specifically that given a natural transformation $\eta : \hom(-,A)\to \hom(-,B)$ (that is, a way to assign a generalized element of $B$ to a generalized element of $A$ in a coherent fashion), there is a unique morphism $f: A\to B$ such that $\eta (g) = f\circ g$ for all $g: X\to A$ generalized element.
The proof is quite easy actually : if you know how to assign generalized elements of $B$ to generalized elements of $A$, just look at "the universal generalized element of $A$" : $id_A: A\to A$. Then you get $\eta_A(id_A) : A\to B$ and that's exactly your $f$ (and the fact that this $f$ works for everyone comes from the naturality of $\eta$)