In Moment, you can set up the program so that every project owner can approve the hours for the co-workers in their own projects. Be aware that hours approval in Moment can only be done by project-owners and not by one specific person.
Go to setup > settings > general > hours approval
In here you can enable "hours approval"
When this is enabled, every time a co-worker logs in hours, the project-owner will receive a notification in the top righ corner in Moment (by clicking on the bell icon)
Project-owners can also go to My pages/Project management/Hours overview and from there approve the hours.