ct.category theory - Equalizer objects in Set.

Given two parallel morphisms $f,g:Xto Y$ in some category $mathcal{C}$, let us consider the category $mathcal{E}_{f,g}$ :

Objects : all pairs $(E,e)$, where $E$ is an object of $mathcal{C}$ and $e:Eto X$ is a morphism in $mathcal{C}$ such that $fcirc e=gcirc e$,

Morphisms : from $(E',e')$ to $(E,e)$ are just morphisms $varphi:E'to E$ in $mathcal{C}$ such that $e'=ecircvarphi$.

Now an equaliser of $f,g$ is just a final object in the category $mathcal{E}_{f,g}$. Final objects in any category are unique (provided they exist), up to a unique morphism; we may then talk of "the" equaliser of $f,g$.

When $mathcal{C}$ is the category of sets, the equaliser always exists (and is therefore uniquely unique); as you say, it is the largest subset of the common source where the two maps coincide.

It is fine to think of it as a "maximal" object in $mathcal{E}=mathcal{E}_{f,g}$, but one must realise that it is also a "minimal" object in the opposite category $mathcal{E}^circ$ in the sense of being an initial object therein.