The subobject classifier in a presheaf category on the walking arrow

On the walking arrow, the arrows into $0$ are only $1_0$, so its sieves are $\mathrm{\varnothing }$ and $\{1_0\}$. The arrows into $1$ are $u$ and $1_1$, and downward closure forces any sieve containing $1_1$ also to contain $u$, giving exactly the three displayed sieves on $1$. Pullback of sieves makes $\mathrm{\Omega }$ a presheaf, and the maximal sieves define the natural truth morphism above.

Consider the representable presheaf $y(1)$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding) and the subpresheaf $S\subset y(1)$ defined by $S(1)=\mathrm{\varnothing }$ and $S(0)=\{u\}$. Its characteristic map $\chi :y(1)\to \mathrm{\Omega }$ sends the unique element $u\in y(1)(0)$ to the maximal sieve $\{1_0\}$, and sends $1_1\in y(1)(1)$ to the sieve $\{u\}$.

For a general subpresheaf $T\subseteq P$, the displayed formula defines a sieve and is natural under restriction. Moreover ${\chi }_{T,c}(x)$ is maximal exactly when $x\in T(c)$: the forward implication tests $1_c$, and the reverse follows from closure of the subpresheaf under restriction. Thus pulling back $\mathrm{true}$ along ${\chi }_T$ recovers $T$, and this condition uniquely determines ${\chi }_T$. By [L1], $\mathrm{\Omega }$ is a subobject classifier.

For the concrete $S\subset y(1)$ of step 2.1, that pullback includes $u$ at object $0$ because ${\chi }_0(u)=\{1_0\}$ is maximal, and excludes $1_1$ at object $1$ because ${\chi }_1(1_1)=\{u\}$ is not maximal. Hence the general classifier calculation gives exactly the displayed example.