Page that links to category content

You can do something like this

 global $post;
 //Get the page name
 $term = get_post( $post )->post_name;

 //If a category with the same name exists, then show the posts from that category
 if( term_exists( $term, 'category') )
 {
     //Show posts from $term category
 }

I hope you get the idea. You can modify it any way to suit your need.