In "Sheaves in Geometry and Logic", M&M define a sieve of an object $C$ as a downward-closed set of arrows $S$ with codomain $C$. They go on to say that for a locally small category, a sieve of $C$ is the same thing as a subfunctor of $\hom(-,C)$, under the rule

`\[S = \{\, f : \exists A . f \in Q(A) \}\]`

Is this always a set for a locally small category? It doesn't seem to be, taking for instance $Q = \hom(-,1)$. Then $S \cong \textrm{ob}(\mathcal C)$, which would imply that $\mathcal C$ is small. Is there something subtle going on here?

setof arrows. However, note that on p36 the category C was introduced as "an arbitrarysmallcategory", so in the context they were working in, a sieve actually is a set. Furthermore, according to their "preliminaries" section, M&M are using the "one universe" foundation according to which even large categories are "sets" (just "large sets") rather than proper classes. – Mike Shulman Sep 8 '10 at 4:19