how to create custom post type with category in Wordpress?

function my_custom_post_product() {
  $args = array();
  register_post_type( 'product', $args );
}
add_action( 'init', 'my_custom_post_product' );

function my_custom_post_products() {
  $labels = array(
    'name'               => _x( 'Products', 'post type general name' ),
    'singular_name'      => _x( 'Product', 'post type singular name' ),
    'add_new'            => _x( 'Add New', 'book' ),
    'add_new_item'       => __( 'Add New Product' ),
    'edit_item'          => __( 'Edit Product' ),
    'new_item'           => __( 'New Product' ),
    'all_items'          => __( 'All Products' ),
    'view_item'          => __( 'View Product' ),
    'search_items'       => __( 'Search Products' ),
    'not_found'          => __( 'No products found' ),
    'not_found_in_trash' => __( 'No products found in the Trash' ),
    'parent_item_colon'  => '',
    'menu_name'          => 'Products'
  );
  $args = array(
    'labels'        => $labels,
    'description'   => 'Holds our products and product specific data',
    'public'        => true,
    'menu_position' => 5,
   'supports'      => array( 'title','author', 'editor','thumbnail','trackbacks' ,'excerpt','comments' ,'custom-fields','revisions','page-attributes','post-formats'),
    'has_archive'   => true,
  );
  register_post_type( 'product', $args );
}
add_action( 'init', 'my_custom_post_products' );


function my_updated_messages( $messages )
{
  global $post, $post_ID;
  $messages['product'] = array(
    0 => '',
    1 => sprintf( __('Product updated. <a href="%s">View product</a>'), esc_url( get_permalink($post_ID) ) ),
    2 => __('Custom field updated.'),
    3 => __('Custom field deleted.'),
    4 => __('Product updated.'),
    5 => isset($_GET['revision']) ? sprintf( __('Product restored to revision from %s'), wp_post_revision_title( (int) $_GET['revision'], false ) ) : false,
    6 => sprintf( __('Product published. <a href="%s">View product</a>'), esc_url( get_permalink($post_ID) ) ),
    7 => __('Product saved.'),
    8 => sprintf( __('Product submitted. <a target="_blank" href="%s">Preview product</a>'), esc_url( add_query_arg( 'preview', 'true', get_permalink($post_ID) ) ) ),
    9 => sprintf( __('Product scheduled for: <strong>%1$s</strong>. <a target="_blank" href="%2$s">Preview product</a>'), date_i18n( __( 'M j, Y @ G:i' ), strtotime( $post->post_date ) ), esc_url( get_permalink($post_ID) ) ),
    10 => sprintf( __('Product draft updated. <a target="_blank" href="%s">Preview product</a>'), esc_url( add_query_arg( 'preview', 'true', get_permalink($post_ID) ) ) ),
  );
  return $messages;
}
add_filter( 'post_updated_messages', 'my_updated_messages' );

function my_taxonomies_product()
{
// Hirarchical false = tags and true = categories
register_taxonomy("product_category", array("product"), array("hierarchical" => true, "label" => "Categories", "singular_label" => "Product", "rewrite" => true));
register_taxonomy("product_tags", array("product"), array("hierarchical" => false, "label" => "Tags", "singular_label" => "tags", "rewrite" => true));
}

add_action( 'init', 'my_taxonomies_product',0);