TITLE: How to give "categorical" specifications of categories like Grp?
QUESTION [9 upvotes]: Certain types of categories (like abelian categories) are specified by listing a set of "categorical" properties that the category must have.   For example, we might demand the category has finite products, or that all monomorphisms are normal.  Sometimes, if you demand enough of these properties, you can uniquely specified a category up to equivalence.  (It might be the one-object, one-morphism category, for instance.)  

Is there any way to list categorical properties that uniquely specify common categories such as $\mathbf{Grp}$, $\mathbf{Vect}$, etc.?

Although this question is not entirely formal (e.g. what does it mean to list "categorical" properties?), I hope it will be clear what type of answer I'm looking for.  I suppose I'd like specifications of categories using "nice" properties, rather than just saying "$\mathbf{Grp}$ is the category whose morphism structure is exactly that given by [groups and group morphisms]."  (I'm aware that $\mathbf{Grp}$ may be defined as the full subcategory of Cat whose objects are groups, but this isn't what I'm looking for.)
I would appreciate a formalization of the question as much as an answer.
Thanks!

REPLY [2 votes]: If you're willing to work with concrete categories, there's a hack that seems to do what you want. Take $\mathbf{Mon}$ for example. We know that:

$\mathbf{Mon}$ is a category.
There's a forgetful functor $U : \mathbf{Set} \leftarrow \mathbf{Mon}$.
There's a natural transformation $\mu:U \Leftarrow U \times U$ that tells us how to multiply the elements of a monoid.
There's a natural transformation $\eta:U \Leftarrow 1$ that tells us what the identity element of each monoid is.
The following hold:

Associativity. For each $X:\mathbf{Mon}$ and all $x,y,z \in UX$, we have $\mu_X(\mu_X(z,y),x) = \mu_X(z,\mu_X(y,x)).$
Unitality. For each $X:\mathbf{Mon}$ and all $x \in UX$, we have $\mu_X(x,\eta(*)) =x$ and $\mu_X(\eta(*),x)$, where $*$ is the unique element of the set $1$.


So by a candidate for the category of monoids, let us mean a category $\mathbf{C}$ together with a forgetful functor $U : \mathbf{Set} \leftarrow \mathbf{C}$ together with two natural transformations $\mu : U \Leftarrow U \times U$ and $\eta : U \Leftarrow 1$, such that the above axioms hold. There's an obvious notion of morphism between such things, and it seems to be the case that $\mathbf{Mon}$ is the terminal candidate for the category of monoids.
This gives us a way of explaining where certain forgetful functors come from. For example, we can view $\mathbf{Ring}$ as a candidate for the category of monoids, so there's a unique morphism $\mathbf{Mon} \leftarrow \mathbf{Ring},$ which is the relevant forgetful functor.