Representable morphism for algebraic spaces I'm trying to understand the definition of algebraic spaces, but there is a notion of representable morphism that is a little confusing to me. Let $S$ be a scheme and let $Sch/S$ denote the category of $S$-schemes. Let $f:F\rightarrow G$ be a morphism of sheaves on $Sch/S$ (with etale topology). We say that $f$ is representable by schemes if for every $S$-scheme $T$ and a morphism $T \rightarrow G$ the fiber product $F\times_G T $ is a scheme.
What exactly are $F$ and $G$? I mean, how can we take the fiber product of a scheme with a sheaf? Also, what is would be the morphism $T \rightarrow G?$

Given $T \in Sch/S$ one obtains a sheaf $h_T = Hom(-,T)$ via the Yoneda embedding. It is common to abuse notation and write $T$ for $h_T$. So a map $T \to G$ means a map $h_T \to G$, i.e. a natural transformation, and $F\times_G T$ means $F\times_G h_T$, which is the fiber product of sheaves. (The fiber product of sheaves can be computed ``pointwise" on objects, i.e. $(F\times_G h_T) (U) = F(U)\times_{G(U)}h_T(U)$.)
