${\bf Set}$ is the category of sheaves on the one-point topological space

Question

It is well-known that ${\bf Set}$ is the category of sheaves on the one-point topological space, i.e. ${\bf Set} \simeq Sh(X,J_{\mathcal{O}(X)})$, with $X=\{x\}$.

Now, the open sets are $\mathcal{O}(X):=\{\emptyset,X\}$. Then $J(\emptyset)=\{ \emptyset,M_\emptyset\}$ (i.e. the empty sieve and the maximal one) and $J(X)=\{M_X\}$.

Now, using matching families, it may be easily shown that any presheaf is a sheaf (unless I did not do the correct computations). My question is: may I construct the equivalence associating with each presheaf $P$ the set $P(X)$?