Wordpress How to add class to <ul> of sidebar widget

The simple, but not so nice and more of a hack, is using Javascript/jQuery.

jQuery( document ).ready(function() {
    jQuery('.widget > ul').addClass('nav nav-list');
});

I really dislike using javascript, but it does the job in this case.

I managed to do it this way:

Sidebar.php:

<!-- sidebar -->
<aside class="sidebar" role="complementary">

<div class="sidebar-widget">
  <?php if(function_exists('dynamic_sidebar')) {

    ob_start();
    dynamic_sidebar('widget-area-1');
    $sidebar = ob_get_contents();
    ob_end_clean();

    $sidebar_corrected_ul = str_replace("<ul>", '<ul class="menu vertical">', $sidebar);

    echo $sidebar_corrected_ul;
    } 

  ?>
</div>

</aside>
<!-- /sidebar -->

I used output buffering to save the sidebar widgets into a variable and then used string replace to find all <ul> tags and replace them with the <ul> tag with the class.