Assume $C$ is a locally small category with equalizers in the sense that given any two arrows $f$ and $g$ with a common source $a$ and target $b$, then there is an object $e$ and an arrow 
$i\colon e\to a$ such that $fi = gi$ that satisfies the usual universal property. Seems that
this generalizes fairly easily (by induction) to any finite subset of the hom-set $C(a,b)$.
That is, given a finite $A\subset C(a,b)$, there is an arrow $i\colon e\to a$ such that
for any $f,g\in A$, we have $fi=gi$ and if $k\colon c\to a$ is such that for any $f,g\in A$
we have $fk=gk$, then there is a unique $h\colon c\to e$ such that $k=ih$.


My question is whether this can be generalized to all of $C(a,b)$ when the hom-set is not
necessarily finite. 



How to generalize equilizers in a category to hom-sets?
Look for "equalizer of a family of arrows" in an arbitrary book on category theory. It is a special case of a limit. A typical example for a category which has finite limits, but not arbitrary limits (and therefore not all infinite equalizers), is the category of schemes.


