Let $X$ be a topological space and denote by $\rm{Op}_X$ the category of open sets with morphisms given by inclusions. Let $\mathcal C$ be a category that admits small limits. When reviewing the gluing axioms for sheaves, it occurred to me that we can rewrite them in a slightly different way. Let me speficy what I mean:
Usually (at least in the references that I am familiar with), given any presheaf $F\colon \rm{Op}_X^{\rm op}\rightarrow \mathcal C$, we say that $F$ is a sheaf if for any $U\in \rm{Op}_X$ and any covering $(U_j)_{j\in J}$ of $U$, the diagram $$F(U)\rightarrow \prod_i F(U_i){{{}\atop {\Large\longrightarrow}}\atop{{\Large\longrightarrow}\atop{}}} \prod_{i,j}F(U_i\cap U_j)$$ is an equalizer diagram. Although this is perhaps closest to the definition given via elements (and thus closest to the intuition coming from gluing functions), I found the following more concise characterization (proof on demand):
A presheaf $F$ as above is a sheaf if for any $U\in \rm{Op}_X$, we find $$F(U)=\lim_{V\subset U}F(V).$$
As I were unable to find this reformulation anywhere, I would like to ask whether this is even correct?