Balanced

Suppose that in a Category we have:
Monomorphism Epimorphism Isomorphism
Then we say that is balanced.

Lemma

If is balanced, then any Separating Family is also a Detecting Family.

Proof

Let be a Separating Family.
Suppose is such that every with
factorizes uniquely through .
Suppose

for some .
Let with
Then

so

because factorizes uniquely through .
This holds for any with
and as is a Separating Family, we conclude

and thus is a Monomorphism.

Now suppose

Any has a unique such that
so we can multiply to find

for any with and thus
so is an Epimorphism.
As is balanced, then is an Isomorphism
and so is a Detecting Family.