Equalizer

Given in a Category
an equalizer of is an Object and map
such that
and for any if
we have a unique
making the Commutative Diagram

Note that is a Monomorphism
and we say that in this case is a Regular Monomorphism

Lemma

If has equalizers, then any Detecting Family in is also a Separating Family.

Proof

Suppose is a Detecting Family.
Let be such that for any with
Let be an equalizer of and .
Then there is a unique such that ,
so factors uniquely through and hence is an Isomorphism.
But so and hence is a Separating Family.