retrieve a menu and format it

Wordpress menu widget

/**
 * MENU widget
 *
 */

class menu_widget extends WP_Widget {

  /**
   * Register widget with WordPress.
   */
  function __construct() {
    parent::__construct(
      'menu-widget', // Base ID
      __( 'Lyra Network - menu widget', 'twentyfifteen' ), // Name
      array( 'description' => __( 'add a menu in the sidebar', 'twentyfifteen' ), ) // Args
    );
  }

  /**
   * Front-end display of widget.
   *
   * @see WP_Widget::widget()
   *
   * @param array $args     Widget arguments.
   * @param array $instance Saved values from database.
   */
  public function widget( $args, $instance ) {
    echo $args['before_widget'];
    if ( ! empty( $instance['menu'] ) ) {
      wp_nav_menu( array(
        'menu_class'     => 'sidebar-nav-menu',
        'menu' => $instance['menu'],
        'menu_id'        => $instance['menu'].'sidebar',
        'menu_class'     => 'side-menu',
        'container' => false,
        'depth'         => 1,
      ) );
    }
    echo $args['after_widget'];
  }

  /**
   * Back-end widget form.
   *
   * @see WP_Widget::form()
   *
   * @param array $instance Previously saved values from database.
   */
  public function form( $instance ) {
    $menu = ! empty( $instance['menu'] ) ? $instance['menu'] : 'main_menu';
    ?>
    <p>
      <label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Please pick a menu:' ); ?></label>
      <?php
      $args  = array(
        'public' => true,
      );
      $post_types = get_post_types( $args, 'names','and' );
      $menus = get_terms( 'nav_menu', array( 'hide_empty' => true ) );

      echo '<select class="widefat"  id="'.$this->get_field_id('menu').'" name="'.$this->get_field_name('menu').' " type="text">';
      foreach ( $menus as $menuObj ) {
        $sel = ( $menuObj->name == $menu ) ? 'selected' : '';
        echo '<option '.$sel.' value="' . $menuObj->name . '" >' . $menuObj->name . '</option>';
      }
      echo '</select>';

      ?>
    </p>
  <?php
  }

  /**
   * Sanitize widget form values as they are saved.
   *
   * @see WP_Widget::update()
   *
   * @param array $new_instance Values just sent to be saved.
   * @param array $old_instance Previously saved values from database.
   *
   * @return array Updated safe values to be saved.
   */
  public function update( $new_instance, $old_instance ) {
    $instance = array();
    $instance['menu'] = ( ! empty( $new_instance['menu'] ) ) ? strip_tags( $new_instance['menu'] ) : '';

    return $instance;
  }

} // class Foo_Widget

// register Foo_Widget widget
function register_menu_widget() {
  register_widget( 'menu_widget' );
}
add_action( 'widgets_init', 'register_menu_widget' );