when a presheaf is a sheaf

Question

I've seen a very natural definition when a presheaf $F:C^{op}\rightarrow Set$ is actually a sheaf. This definition used the functors $hom(-,-)$ and $F$ and notions of injective and surjective maps between two particular sets of natural transformations and probably also the notion of subfunctor.It claimed that the induced map,whose definition I do not remember now,and which I would like to understand, is always injective and it is surjective iff $F$ is a sheaf.I'm unable to Google this definition again. Any help or hint how such a definition may go?

Answer 1

$F$ is a sheaf if and only if the map $$\mathrm{Hom} (h_X, F) \to \mathrm{Hom} (\mathfrak{U}, F)$$ is a bijection for all covering sieves $\mathfrak{U} \hookrightarrow h_X$. This is just a sophisticated way of rephrasing the standard definition using equalisers and products.