So he has explicitly created derivative work of Postgres, that's not the question at all. The question is whether the conditions of the AGPL for distributing the source would trigger in his case. I would say yes but IANAL.
Just because I create a derivative work of an (A)GPL project does not immediately require me to publish those changes. Only if I distribute the work, I also need to distribute the changes along with it.
While a derivative work of Postgres is created, it is not "distributed" (in the sense of the AGPL). The web application is, which I do not consider a derivative work of Postgres in the general case.
If you implement a feature in Postgres and have a web application that is practically just a wrapper around this one, new feature, then that could be considered a derivative work. But I'd say that's a far stretch.
"... and I modify it to my needs ..."
So he has explicitly created derivative work of Postgres, that's not the question at all. The question is whether the conditions of the AGPL for distributing the source would trigger in his case. I would say yes but IANAL.