Start with both.
predeterminer, determiner, pronoun
(referring to) two people or things together:
Both my parents are teachers.
They have two children, both of whom live abroad.
She has written two novels, both of which have been made into television series.
In all these examples, both may be removed with loss of emphasis but with no loss of meaning:
My parents are teachers
They have two children who live abroad
She has written two novels, which have been made into television series.
In this usage, the determiner emphasises the commonality of the feature: teaching, living abroad, television series. It points to a feature that each person may or may may not necessarily have.
Things are a little different with your example.
with : together : joint : jointly
In this meaning of co-, the feature of existing is necessarily shared, is joint. The addition of both adds no extra emphasis and is unnecessary.
In conclusion, your co-author is advocating an acceptable and common usage that is not wrong, merely redundant in your example, but your own position is concise, justified and defensible.