Exchange 2013 Set Outlook using the internal host name

Exchange 2013 certificate change - Outlook connection problem

Good day,

I wanted to connect a running Exchange Server with a new certificate. So far it was done through a certificate issued by Exchange. Now I have a public one
Certificate created which, however, no longer contains the local Exchange name. Now Outlook always complains when starting with the following message.
The certificate is integrated in the IIS. External OWA works without a certificate notification.


There is a problem with the proxy server's security certificate.
The name of the security certificate is invalid or does not match the name of the target website `exchange.domain.local
Outlook cannot connect to the proxy server (error code 10)

Now my question, where do I have to change the name in Exchange so that the new certificate is also recognized by Outlook and this error message no longer appears.
What exactly is Outlook addressing?

Unfortunately I don't know what to do next.
I appreciate help.

greeting
Andreas