For example, deciding whether or not the following is a category seems to depend on the above question (from Awodey's Category Theory, pg. 6):

"What if we take sets as objects and as arrows, those $f : A \rightarrow B$ such that for all $b \in B$, the subset $f$^{-1}$(b) \subseteq A$ is finite?"

Define for each $n \in N$ the function $f$_{n}: $N \rightarrow N$ where $f$_{n}$(x) = max(0, x - n)$.

Then any $f$_{n} or any finite composition thereof has finite inverse images. Yet the "infinite composition" $... f$_{1}$f$_{2}$f$_{0} has an infinite inverse image for 0, and so the above does not meet the definition of a category.

If this "infinite composition" is legit, does it follow from the basic definition of a category, or must the definition be made more flexible or precise to accommodate it?

For reference, this is Awodey's definition concerning composition:

"Given arrows $f : A \rightarrow B$ and $g : B \rightarrow C$, [...] there is given an arrow: $g$ o $f : A \rightarrow C$ called the composite of $f$ and $g$."

Thank you for your insight.

couldask for ways to modify the usual definition of categories so as to be able to attach some sense to (some, at least) infinite compositions, as that is a question which has answers. One straightforward way is to consider categories whose End-sets are topological spaces (for example, categories enriched over topological spaces), where you can make sense of what the limit of a sequence of endomorphisms of a fixed object is. – Mariano Suárez-Alvarez♦ Apr 29 '10 at 5:38