TITLE: What's a reasonable category that is not locally small?
QUESTION [37 upvotes]: Recall that a category C is small if the class of its morphisms is a set; otherwise, it is large.  One of many examples of a large category is Set, for Russell's paradox reasons.  A category C is locally small if the class of morphisms between any two of its objects is a set.  Of course, a small category is necessarily locally small.  The converse is not true, as Set is a counterexample.
Now, I can construct categories that are not locally small.  However, what's the most common or most reasonable such category?

REPLY [9 votes]: The category of Grothendieck topoi and (equivalence classes of) geometric morphisms. For example, if $A$ is the classifying topos for abelian groups, the geometric morphisms from $Set$ to $A$ are in bijection with isomorphism classes of abelian groups, which is certainly not a set.