Arrows in a presheaf category covariantly induce maps between subfunctors

Question

I am currently reading Sheaves in Geometry and Logic and have hit a little bit of a snag that I can't figure out. In the book it is claimed that if $ g\colon B\longrightarrow C $ is morphism and $Q$ is a subfunctor of $Hom(\_,C)$ then this determines a subfunctor $Q'$ of $Hom(\_,B)$. The claim is that $g$ induces $Q'$ from $Q$ "by pullback". I am at a loss as to how this is induced. We want $Q'(D)$ to be a subset of $Hom(D,B)$ for any object $D$ and $Q(f:D\rightarrow D')$ to be the restriction of $Hom(f,B)$ for any arrow $f$, but I have no idea how to use $Q'$ and $g$ to create this new functor.

Answer 1

The definition would be $$ Q'(D) = \{f: D \to B \mid gf \in Q(D)\}. $$ we can easily check that $Q'(h: D \to D'): Q'(D') \to Q'(D)$ is the restriction of $\operatorname{Hom}(h, B)$ . Let $f \in Q'(D')$, then $gf \in Q(D')$, so $gfh \in Q(D)$. This follows because $Q$ is a subfunctor of $\operatorname{Hom}(-, C)$. Hence $fh \in Q'(D)$, as required.