The answer to this is probably simple, but it eludes me.
Here's the scenario: At the moment, I have a server with a certificate issued to it using HTTPS that a customer POSTs responses to after processing requests sent from a web service. The customer's server has their own certificate issued to it and it also uses HTTPS. Neither certificate is a wildcard cert.
As part of the original setup, the customer requested that my server had a certificate (unsurprisingly). Now, the customer has an additional requirement as part of a recent upgrade. They want my root cert installed on their server.
As I mentioned above, my server already has a valid, non-wildcard certificate. When their server POSTs to mine, they already see that valid cert, and when my server talks to their server I see their cert. So what would my customer gain from this?
Edit: To clarify, my certificate is signed by a third party CA.