Do I need to open their web application code when using Berkeley DB (released under the GNU AGPL)?

Question

Licenses for software and other works

Let's say I have a web application, for example, some site that uses Berkeley DB as a database.
Berkeley DB itself is released under the GNU AGPL, which has a special clause stating that users who interact with the program over a network must also be able to get the source codes.
Does this mean that I will be required to provide any user of the site with its source code?

PS If it doesn't make it difficult, not just "yes / no", but why exactly.