URL redirection?

URL redirection also called URL forwarding.

The technology makes a web page available under more than one URL address.

ex) When a browser attempts to open a URL that has been redirected, a page with a different URL is opened.

Similarly, domain redirection or domain forwarding is when all pages in a URL domain are redirected to a different domain, as when wikipedia.com and wikipedia.net are automatically redirected to wikipedia.org

Applying the concept, when users visit the website through own mobile, website may redirect to the website on the mobile version.

Reference 

https://en.wikipedia.org/wiki/URL_redirection