Taxonomy Category And Subcategory

Hello guys, I’m not very good at English, but I’m gonna try.

I’m in the following case:

– I need to create a panel session to register products with their respective categories and subcategories.

EX:
Category:
– Vase / Chair / T-shirt

SubCategory:
– Vase
— Clay Pots (Pot belongs to category Vase)
— Vessels Cement (belongs to category Vase)

– Chair
– Reclining Chairs (Chair belongs to category Chair)
– Padded Chairs (Chair belongs to category Chair)

– T-shirt
– Regatta (belongs to category T-shirt)
– Polo (belongs to category T-shirt)

I need to create this structure to register a product.

EX:

Product: ProdX
– Category: T-shirt
– Subcategory: Polo

Well, I do not know how I do the record straight, I only have part of the product / category ready.

add_action('init', 'product_register');
if (!function_exists('product_register')) {
    function product_register() {
        $labels = array(
            'name' => __('Product', 'tkwave'),
            'singular_name' => __('Product', 'tkwave'),
            'add_new' => __('Add New', 'tkwave'),
            'add_new_item' => __('Add New Product', 'tkwave'),
            'edit_item' => __('Edit Product', 'tkwave'),
            'new_item' => __('New Product', 'tkwave'),
            'all_items' => __('All Products', 'tkwave'),
            'view_item' => __('View Product', 'tkwave'),
            'search_items' => __('Search Products', 'tkwave'),
            'not_found' =>  __('No Producct found', 'tkwave'),
            'not_found_in_trash' => __('No Product found in Trash', 'tkwave'),
            'menu_name' => __('Products', 'tkwave')
        );
        $args = array(
            'labels' => $labels,
            'public' => true,
            '_builtin' => false,
            'capability_type' => 'post',
            'menu_icon' => THEME_DIR . '/framework/assets/images/product.png',
            'hierarchical' => false,
            'rewrite' => array( 'slug' => 'product'),
            'supports' => array('title', 'editor', 'thumbnail')
        );
        register_post_type('product', $args);
        flush_rewrite_rules();
    }
}

register_taxonomy("products", array("product"), array("hierarchical" => true, "label" => __("Categories", "tkwave"), "singular_label" => __("Product Categorie", "tkwave"), "rewrite" => true));