The opposite thing to the ringed space

Question

It is common that a sheaf of rings is a sheaf (of sets) $\cal F$ on a space $X$ such that for each open $U\subseteq X$, ${\cal F}(U)$ has the structure of a ring and for each open $V \subseteq U$ , the restriction map ${\cal F}(U) \to {\cal F} (V)$ is a ring homomorphism. My question is this: Can we naturally obtain the opposite thing: for each open $V \subseteq U$ (a/the?) map ${\cal F}(V) \to {\cal F} (U)$ is a ring homomorphism. I think that there is some obstacle in general, but I'm not sure about its nature.