Nomrally the search works with the search widget placed in a sidebar. What if you want to show the product search elsewere?
Now you can show the product search anywhere in the website content with [[product_search]] shortcode. You can place the shortcode on any page, blog post, product page or any other WordPress content.
You can even place the search shortcode between paragraphs. See the example output:
The layout will be different for different themes as the search form is always adjusted to match the currently used WordPress theme.