Arrows in a presheaf category covariantly induce maps between subfunctors

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 $$gcolon Blongrightarrow C$$ 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 $$Q(f:Drightarrow D’)$$ 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.

Mathematics Asked by subobject_classifier on January 2, 2021

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.

Correct answer by Mark Kamsma on January 2, 2021

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