TITLE: Axiom of Replacement in Category Theory
QUESTION [12 upvotes]: Does the usual development of category theory (within Goedel-Bernays set theory, for example) require the axiom of replacement?  I would have asserted that this was obviously true, but it seems to be common wisdom that the axiom of replacement is an exotic axiom not used outside of axiomatic set theory.  Also, sadly the overlap between things that I have thought were obviously true and were in fact false is larger than I would like to admit...
The axiom of replacement basically says that if a class is the same size as a set, then it is a set.  This allows us to identify classes that are sets as being small and those that are proper classes as large.  Without replacement, you could have countable classes that not sets.
I don't see how to construct most limits and colimits in familiar categories such as Set or Top without the use of replacement.  Already, for an infinite set $X$, I don't see how you construct 
$$
\coprod_{i=0}^\infty P^i X,
$$
where $P^i$ is the $i$-th power set of $X$ (i.e. $P^0 X = X$ and $P^i X = P (P^{i-1}X$)).
Using replacement, it's easy to construct.  You form the set 
$$
\bigcup_{i=0}^\infty P^i X,
$$
and then the coproduct is isomorphic to the set of pairs $(i, x)$, where $x \in P^i X$.  Am I completely misunderstanding the issue here, and this coproduct can be proven to exist without replacement?
I suppose you could cook up a definition of diagram so that in the absence of replacement, you cannot even form the diagram for this particular coproduct, but that would be sort-of unsatisfying.

REPLY [2 votes]: The other answers and comments so far indicate that the axiom of replacement is needed to show the existence of coproducts in Set. This is not true if we adjust the definitions properly ;).
The existence of coproducts in a category $C$ may be formulated that if $F$ is a set of objects in $C$, then there is a object $\coprod F$ with the usual universal property. Note that the usual formulation uses a function from a set to the class of objects of $C$ (a "family"), which almost always requires an application of the axiom of replacement, which is somehow artificial.
Now in the category of sets, let $F$ be a set of sets. Then $x \mapsto x \times \{x \}$ is a function $F \to F \times P(F)$. By comprehension, the image exists and this will be $\coprod F$.
I believe that most of (all?) category theory can be developed with definitions as above and then we do not have to use replacement to show the standard category theoretic properties of the categories in practice.
There is a nice exercise in Kunen's Set theory (IV, Ex. 9), in which the reader has to develope 99 % of modern mathematicss in ZC, which is ZFC without replacement ;-).