I encountered the following situation:


Let C be an abelian category and X, Y be objects of C verifying
$
Hom\_C(X, Z)\cong Hom\_C(Y, Z)
$
For any object Z of C.


Does it follow that $ X \cong Y$?


My belief is that in general this not true. The real question is what additional hypothesis have to be satisfied by C in order to obtain isomorphic objects?


For example, if C closed then probably these two objects are isomorphic.



A question in category theory
Let $C$ be a category, and $X$ an object in $C$. Then $h\_{X} = \textrm{Hom}(X, \\_)$ is a functor from $C$ to $\textrm{Set}$.


If for two objects $X,Y$ in $C$ the functors $h\_{X}$ and $h\_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.


In fancy categorical terms it says that the functor $h \colon C^{\textrm{opp}} \to \textrm{Func}(C, \textrm{Set})$ is fully faithful. The slogan is *"tell me who your friends are, and I will tell you who you are"*. So if you know all the arrows from (or to) an object, then you can determine the object up to isomorphism.


