Definition of a sieve and hom functors On page 37 of Sheaves in Geometry and Logic, it is mentioned:

Now if $Q \subset Hom_{C}(-,C)$ is a subfunctor, the set
$$S = \{f | \text{ for some object A, } f: A \rightarrow C \text{ and } f \in Q(A) \}$$
is clearly a sieve.

Now according to my understanding, the set $Q(A)$ is a subset of $Hom(A,C)$ thus it contains arrows of the form $A \longrightarrow C$. Now a sieve is a downwards closed set, thus if we have any other arrow $h$ and we compose it with an arrow in $S$ we should obtain another element in $S$.
Suppose we take $h: B \rightarrow A$, then we have that $fh : B \rightarrow C$, but this arrow has domain $B$ so it doesn't belong to $S$.
What am I misunderstanding here?
 A: Let’s write the definitions:

*

*Given a locally small category $\newcommand{\C}{\mathbf{C}} \C$, a subfunctor of $P \colon \C^{\rm op} \to \mathbf{Set}$ is a functor $Q \colon \C^{\rm op} \to \mathbf{Set}$ such that:

*

*$Q(A) \subseteq P(A)$ for every $A \in \C$; and

*if $h \colon B \to A$ is an arrow of $\C$, then $P(h) \colon P(A) \to P(B)$ maps $Q(A)$ into $Q(B)$, and $Q(h) \colon Q(A) \to Q(B)$ is the restriction of $P(h)$.



*Given an object $C$ in a category $\C$, a sieve on $C$ is a set $S$ of arrows of $\C$ with codomain $C$ such that if $f \colon A \to C$ is in $S$, then for every arrow $h \colon B \to A$, $fh$ is in $S$ too.
Now, if $Q$ is a subfunctor of $\newcommand{\Hom}{\text{Hom}} \Hom_\C(\_,C)$, we claim that $S := \bigcup_{A \in \C} Q(A)$ is a sieve on $C$.
Indeed, if $f \in S$, then $f \in Q(A)$ for some $A \in \C$; so, given an arrow $h \colon B \to A$, we have that $fh = \Hom_\C(h,C)(f) \in Q(B) \subseteq S$ because $$\Hom_\C(h,C) \colon \Hom_\C(A,C) \to \Hom_\C(B,C)$$ maps $Q(A)$ into $Q(B)$.
