Recovering an object from its category Consider the category of groups (but the question arises for any category of mathematical object, basically).  It is easy to read off what the automorphism group of a group is or what its cardinality is (from the number of morphisms from the two element group), 

but how can I determine what its internal automorphism group is or, informally, how can I determine what group it really is?  

Or, if that is not precise, given a property of groups such that I can write a sentence in higher order logic (or maybe some even more powerful logic) that will be true in just those groups that have the property, how can I be sure that there is a formula of higher order logic (or the more powerful logic) that will be true of just those members of the category of groups that have the property? 
And if there is no general guarantee that this is possible in the case of groups or in the case of some other kind of mathematical object, doesn't that somehow mean that the category of that kind of mathematical object isn't really capturing everything about them?
 A: In the particular case of groups, you can recover the underlying set of a group $G$ as the Hom-set $\text{Hom}(\mathbb{Z}, G)$. In other words, $\mathbb{Z}$ represents the forgetful functor $\text{Grp} \to \text{Set}$. You can recover the group operations on $G$ by endowing $\mathbb{Z}$ with the structure of a cogroup object; see this blog post for the details. From here you can do everything else. 
In general, one important caveat to keep in mind is that categories are not automatically concrete; that is, they do not automatically come with a forgetful functor $C \to \text{Set}$. Some properties look like they're properties of objects of $C$ but are actually properties of objects of $C$ together with a choice of such a functor (e.g. "cardinality of the underlying set"), and in general such functors neither exist nor are unique. If you want to talk about such properties, in some sense you need to provide the functor as extra data (but this is debatable; in another sense the functor is part of the correct definition of the property.)
But this is a blessing, not a curse: working in the language of category theory forces you to recognize what properties do and do not depend on a choice of concretization, in the same way that working in the language of group theory forces you to recognize what properties do and do not depend on a choice of group action. This is a valuable exercise for understanding categories that don't have a good choice of concretization (e.g. the category of schemes.) 
Another important caveat to keep in mind is that when putting an object in a particular category $C$, you can only hope to recover the object up to isomorphism in $C$. Many familiar mathematical objects are members of lots of different categories. For example, $\mathbb{R}$ is


*

*a group

*a ring

*a topological space

*a totally ordered set

*a metric space

*a manifold

*a smooth manifold

*a topological group

*a Lie group


and so forth, and putting it in any of the above categories will only let you recover it up to isomorphism in the appropriate senses. 
