To share a taxonomy between multiple post types when you register her, you have to pass an array with the slug if all posts types that she will belong to.
Example:
register_taxonomy( 'category', array( 'work', 'another_cpt' ), $args );
But in your case the category taxonomy is already registered, so you have to do the following:
function add_category_to_work_cpt(){
register_taxonomy_for_object_type( 'category', 'work' );
}
add_action( 'init', 'add_category_to_work_cpt' );