Problem with Awodey's Category Theory 8.8 Topoi, an alternative characterization of subobject classifiers.

Question

Definition 8.16. Let $\mathcal{E}$ be a category with all finite limits. A subobject classifier in $\mathcal{E}$ consists of an object $\Omega$ together with an arrow $t : 1 \to \Omega$ that is a "universal subobject," in the following sence:

Given any object $E$ and any subobject $U \to E$ (injective, but mse doesn't support tikzcd), there is a unique arrow $u: E \to \Omega$ making the following diagram a pullback:

$\cdots$

It is easy to show that a subobject classifier is unique up to isomorphism: the pullback condition is clearly equivalent to requiring the contravariant subobject functor, $$\mathrm{Sub}_{\mathcal{E}}(-): \mathcal{E}^{op} \to \mathbb{Sets}$$ (which acts by pullback) to be representable, $$\mathrm{Sub}_{\mathcal{E}}(-) \cong \mathrm{Hom}_{\mathcal{E}}(-, \Omega)$$

It is easy to go left to right, but I'm having trouble going the other way, from the isomorphism of the two functors to the desired UMP.

Thanks.

Answer 1

This is the usual argument how to recover a universal property from a representable presheaf. Plugging in $\Omega$ into the isomorphism of functors yields a universal subobject $t$ that corresponds to the identity on $\Omega$, and then the desired universality follows immediately since every map into $f\colon E\to\Omega$ is the image of the identity on $\Omega$ along $f^\ast$ and since $f^\ast$ acts on subobjects by means of pullback along $f$.

Edit: To see that the domain $X$ of $t$ is the terminal object, note that any morphism $f\colon E\to X$ determines a map $tf$ and therefore a subobject of $E$. By construction, $E$ must be a retract of this subobject, which is only possible if the subobject is the maximal one (i.e. the identity on $E$). As the identity on $E$ defines a subobject of $E$ and therefore gives rise to such a map $f$, this shows that there is a unique such map $f$, hence $X$ is terminal.