My custom posts that are assigned to categories, are not called on a category search using a Category Menu

Category archive by default it search for posts and not your custom type, you need to tell WordPress to search for your custom type, paste this code in your theme’s functions.php file:

function cpt_Search_category_Filter($query) {
    $post_type = array('post','business_sold');
    if ($query->is_search || $query->is_category) {
        $query->set('post_type', $post_type);
    };
    return $query;
};

add_filter('pre_get_posts','cpt_Search_category_Filter');

and you should be fine.