The plus construction turns a separated presheaf into a sheaf

Let S consist of arrows h: Z → X for which there are f: Y → X in R, g: Z → Y with h = fg, and a representative (T,s) of $x_f$ with g in T. Assign h the value $s_g$ in F(Z). This does not depend on the factorization or representative: the matching condition for the $x_f$ and the definition of equality in F+ make any two proposed values locally equal after a covering pullback, and separatedness of F makes them equal. Restrictions preserve the assigned value, so these values form a matching family of F on the sieve S.

For each f in R, a representative sieve T of $x_f$ is contained in $f^{\ast }S$, so $f^{\ast }S$ covers. Transitivity makes S covering. Its matching family determines a class x in F+(X). For each f in R, the restriction of x and $x_f$ agree on the covering sieve T, so x restricts to $x_f$. Separatedness of F+ gives uniqueness. Thus F+ is a sheaf.