Let $\mathbf{Cat}$ be the category of small categories.
(1) Is $\mathbf{Cat}$ complete?
(2) Is $\mathbf{Cat}$ cocomplete?
Remark(Jan. 11, 2013)
Of course, the question is implicitly asking for a proof if the answer is affirmative.
It's easy to see that $\mathbf{Cat}$ has products and equalizers.
Hence it is complete.
However, I'm not sure it is cocomplete.
Since it clearly has coproducts, the question is reduced to whether it has coequalizers or not.
Let $F, G \colon \mathcal{C} \rightarrow \mathcal{D}$ be functors.
Ittay Weiss wrote that a coequalizer is the quotient of $\mathcal{D}$ by the congruence generated by equating $F(X)$ with $G(X)$.
It is easy to define the congruence on $Ob(\mathcal{D})$ generated by equating $F(X)$ with $G(X)$.
However, I have no idea how to define the congruence on $Mor(\mathcal{D})$ generated by equating $F(X)$ with $G(X)$.
Edit(Jan. 17, 2013)
To the downvoters, why don't you reset your votes? The question is clearly important in category theory. I know you don't like me, but the question has nothing to do with your liking towards me. I'm saying this not because I care reps, but because the negative votes are sending wrong signals to the users.
Best Answer
$Cat$ is both small complete and small cocomplete. It is not large complete nor large cocomplete since it is not a poset.
The only non-trivial part of the proof of small completeness and small cocompleteness is the construction of coequalizers. An explicit, and rather elementary, construction of coequalizers is given in the article generalized congruences; epis in Cat, TAC 1999. The notion of coequalizer (as well as that of epi) in the category of small categories was proved to be non-elementary (in a precise logic theoretic meaning) by John Isbell in 1968 in the article "Epimorphisms and dominions III".