You need to include the QSA flag at the end of your rule. This appends any existing query string to the one that you're constructing in the rule itself:
RewriteEngine On
RewriteRule ^category/([0-9]+)/[a-z][-a-z0-9]*$ cat.php?id=$1 [L,QSA]
So the rewritten URI should end up being: /cat.php?id=23&page=3