We usually treat an equalizer as, for example, $(E, e)$ -- where it is understood that $e$ is an arrow from $E$. So we wouldn't lose information if we just identified the equalizer with $e$, given the arrow determines its source.
We usually identify a subobject $S$ of $X$ with a (monic) arrow $s$ from $S$ to $X$. But we could have treated it officially as a pair $(S, s)$, i.e. $S$ equipped with $s$.
And with e.g. the objects in slice categories, people go either way. Some treat the objects of $C/X$ explicitly as pairs $(A, f)$ for any ${C}$-object $A$ and ${C}$-arrow $f\colon A \to X$. Others take the objects just to be $C$ arrows, since the arrow determines its source.
Now, I confess I've always taken it (without much thought) that whether in the various cases we officially specify an arrow or an arrow-with-its-source is a matter of tradition and/or presentational felicity. But is that right? Is there, for example, any slightly deeper reason why we officially present a subobject of $X$ as a monic into $X$ rather than an object $S$ equipped with a monic into $X$?