Let $\mathbb{C}$ be a small category and Psh($\mathbb{C}$) the category of presheaves over $\mathbb{C}$. I just want to consider representable presheaves.
The funtor Sub maps a representable presheaf $h_X=\hom(-,X)$ in the subfunctor $S$ that has codomain $X$ and with the property that if $f\in S$ and for some $h$ $fh$ is defined then $fh\in S$. Is this right? Can I write it in a better way?
I also know that a subfunctor $S$ of a functor $Q$ is a class of monos, which is a class of natural transformations that are injective on every slice.
So the question is: I know that Sub maps arrows into pullbacks, but I can't understand in this case where do the natural transformation between representable presheaves go? Which is the pullback? I can do this in Set, but having functors and natural transformation I'm in trouble.
I just need to know: if $f:A\rightarrow B$ is a map between representables, what is $Sub(f)=f^*: Sub(B)\rightarrow Sub(A)$?
Thank you so much!