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.