Definition
(Presheaf). A presheaf on a category \(\mathcal S\) is a contravariant functor \(\mathcal S\to \text{Sets}\).
(Sheaf). A sheaf on a site \(\mathcal S\) is a presheaf \(F:\mathcal S\to \text{Sets}\) such that for every object \(S\in \mathcal S\) and covering \(\{S_i \to S\}\in \text{Cov}(S)\), the sequence \[F(S) \to \prod_iF(S_i) \rightrightarrows \prod_{i,j} F(S_i\times_S S_j)\] is exact, where the two maps \(F(S_i) \rightrightarrows F(S_i\times_S S_j)\) are induced by the two maps \(S_i\times_S S_j\to S_i\) and \(S_i\times_S S_j \to S_j\).