How does equalizer (category theory) characterize set theory equalizer? In the category Set, the equalizer of $f: A \to B$ is the (largest) set of elements $x \in A$ such that $f(x) = g(x)$. But in category theory, this is generalized to a map $E \xrightarrow{e} A$ such that $f\circ e = g\circ e$ and such that the UMP holds. But looking back at the set example, can't you take a smaller subset than the largest and surely that would (the inclusion) be a map $e$. In other words, how does being an equalizer ensure that $E$ is indeed the "largest" object?

Let $e:E\to A$ be a categorical equalizer of $f,g:A\to B$, and let $D$ be the set-theoretical equalizer. What we want to show is that the image of $e$ is $D$.

To see this, let $d:D\hookrightarrow A$ be the obvious inclusion. Since $f\circ d=g\circ d$, by the universal property of $e$, $d$ must factor through it - there is some $c:D\to E$ such that $d=e\circ c$. Since $d$ is onto $D$, so must be $e$.