In SHEAVES, COSHEAVES AND APPLICATIONS, the author introduced a different definition for sheaves. Namely,
Definition: X is a topological space and $\mathcal{D}$ is a category, e.g. $\mathbf{Sets}$, $\mathbf{Vect}$, e.t.c.. Suppose $\mathcal{F}:\mathbf{Open}(X)^{op}\rightarrow \mathcal{D}$ is a presheaf, $\mathcal{U}=\{U_i\}$ is an open cover of a open set $U\subset X$, we say that $\mathcal{F}$ is a sheaf on $U$ if the unique map from $\mathcal{F}(U)$ to the limit of $\mathcal{F}\circ \iota^{op}_\mathcal{U}$ is an isomorphism. Here $\iota_\mathcal{U}$ is the canonical functor from nerve $\mathcal{N}(\mathcal{U})$ to $\mathbf{Open}(X)$.
In section 2.2, the author left an exercise to prove this definition is equivalent to the sheaf axioms of locality and gluing (see Sheaf (mathematics)). He implies that we should utilize a structure theorem in category theory: limits can be expressed by products and equalizers (Theorem 2.2.1).
However, I haven’t made any progress following this idea. The author said that “ Limit or colimit over the nerve of a cover can be determined after considering only the elements of the cover and their pairwise intersections. Do this by observing that the limit or colimit over the 1-skeleton of the nerve defines a cone or cocone over the whole nerve and employing universal properties.” To start with, is it possible to prove $\prod \mathcal{F}(U_i)\cong \prod \mathcal{F}(U_I)$, where $I$ corresponds to all objects in $\mathcal{N}(\mathcal{U})$?