The following is proposition 1 from Maclane & Moerdijk's Sheaves in Geometry and Logic, part II, section 1.
Proposition 1. If $F$ is a sheaf on $X$, then a subfunctor $S\subset F$ is a subsheaf if and only if, for every open set $U$ and every element $f\in FU$, and every open covering $U=\bigcup U_i$, one has $f\in SU$ if and only iff $f|_{U_i}\in SU_i$ for all $i$.
I really feel i'm missing the story here. Why, generally speaking, isn't the condition [$f\in SU$ if and only iff $f|_{U_i}\in SU_i$ for all $i$] enough for a general presheaf, not necessarily a subfunctor of a sheaf, $S$ to be a sheaf?