As you can see in this action, it is overwriting all existing tags by catname, how to add a new tag, preserving the current tags, and not only in update but in publish event, since I use an import plugin, And currently I have to enter all posts and click on update to take the desired action.
add_filter('wp_insert_post', 'add_cat_to_tags', 10, 3 );
function add_cat_to_tags( $post_ID, $post, $update ) {
$tags = array();
$cats = get_the_category( $post_ID );
foreach ( $cats as $cat ) {
$tags[] .= $cat->name;
}
// overwrites any existing tags!!
wp_set_post_tags( $post_ID, $tags );
}
Thanks all.