Your certificate is a text file which is publicly available, for example you can look it up in a service like crt.sh or you can save it from your web browser when visiting the site.
Your certificate can’t be used by a server to identify itself unless the server also possesses the corresponding private key. That’s why I, for example, can’t just save the certificate from
www.google.com and install it onto my own web server and then convince people that I’m Google!
If you want to use the certificate on a separate site, you’ll have to copy the private key, chain (intermediate) certificate, and certificate from the original server onto the new server. These files are found under
/etc/letsencrypt if you used Certbot. You can copy them by using
rsync, for example.
Certbot on the new server wouldn’t be able to do this for you on its own, because there’s no way that it can access or obtain the private key!
A difficulty with this is that when the certificate is replaced with a new version (prior to its expiry), the new files also need to be copied onto every server that’s going to use them. To help automate this, we’ve provided a feature in Certbot called
--deploy-hook which lets you specify a script or command of your choice to run that copies the new certificates to wherever they need to be copied (e.g., onto any other servers that use the same certificate and key).