Small categories and completeness (1) Can a small category be cocomplete? Meaning, have all small colimits? I'd be glad to see an example.
(2) Suppose $\mathcal C$ is a small category, with $Ob(\mathcal C)$ being of cardinality $\kappa$. May $\mathcal C$ have all small limits of cardinality $\leq \kappa$ ?
Does this now allow examples which ware outruled in (1)?
 A: Small (co)complete categories are posets by a theorem of Freyd.
If $C$ has all small coproducts and its class of morphisms $C_1$ is small, then $C(x,y)^{C_1}\simeq C(\coprod_{f\in C_1} x, y)\subseteq C_1$. If $C(x,y)>1$, then $C_1$ has a subset of strictly greater cardinality: contradiction.
A poset that has suprema and infima of all of its subsets is a complete category.
A: (1) Yes, the trivial (final) category, with only one object and one morphism (the identity in the unique object).
(2) If you only asked for colimits of cardinality $<\kappa$ then you would find many examples, e.g. finite abelian groups ($\kappa = \aleph_0$ here). If you insists in $\leq\kappa$ I think you face the same problem as above. Just think of vector spaces of dimension $<\kappa$. This category, up to isomorphism, has $\leq \kappa$ objects but it doesn't have colimits of sice $\leq \kappa$ since the coproduct of $\kappa$ copies of the ground field has dimension exactly $\kappa$.
I've been speaking about colimits in (2) instead of limits, which is what you ask for, so take opposite categories.
