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.