I'll start with example:

Let $X$ be a scheme, and $O_X$ be its structure sheaf. It is defined at the moment on open sets of $X$, and it takes them to $Sets$. However, it is extendable to a sheaf on the Zariski site of $Sch$ by: Take a scheme $S$ to $\mathbb{G}_a(S)$. Now that it is a functor $Sch \rightarrow Sets$, it makes sense to ask whether it is representable, which in this case it is (by $\mathbb{Z}[X]$), and it is even a group scheme.

My, somewhat vague, question is: how prevalent is the phenomenon? For example, are all coherent sheaves on any scheme extendable to representable functors? To group schemes? Is there an iff condition for this to happen?