WooCommerce - Insert category name(s) before product

<?php
  // insert category name(s) before product in shop loop
  // products can have none, one or many categories
  // code goes in functions.php for your child theme
  add_action('woocommerce_before_shop_loop_item', 'insert_cat_names');
  function insert_cat_names() {
    $terms = get_the_terms($product->ID, 'product_cat');
    $nr_terms = count($terms);
    if ($nr_terms == 1) {
      print '<p class="cat_names">Category: '.$terms[0]->name.'</p>';
    }
    if ($nr_terms > 1) {
      print '<p class="cat_names">Categories: ';
      $first = true;
      foreach ($terms as $term) {
        if ($first) {
          $first = false;
        } else {
          print ', ';
        }
        print $term->name;
      }
      print '</p>';
    }
  }