Definition. J-sheaf [ZUQN]

Let $(\mathcal{C}, J)$ be a site, a $J$-sheaf on $\mathcal{C}$ is a presheaf $P : \mathcal{C}^{op} \to \text{Set}$ such that for each $J$-covering sieve $S \mathbin{⪧}_J X$ and for each family

such that for each $f \in S$ and for each morphism $g$ that's $\mathcal{C}$-composable with $f$

, there exists a unique element $x \in P(X)$ such that

for all $f \in S$.