Using the category_url_title as the category link

Hey there, the subject might be a little vague, let me explain:

<a href="https://expressionengine.com/products">{category_name}</a>

Generates this URL:

http://test.dev/index.php/products/category/test-category

However, what I’d like to do is use the {category_url_title} as the second segment instead of category, so it would look like this:

http://test.dev/index.php/products/test-category

I’d also like to achieve this with single entries, as well:

http://test.dev/index.php/products/product_url_title

My structure is basically this:

| product.group
| - - index.html

| products.group
| - - index.html
| - - category.html

But I’d like to group all of the above in the products.group template group if possible, so that I could have a structure like this, for example:

| products.group
| - - index.html
| - - category.html
| - - product.html

Without having to use the /category/ or /product/ segments in the URL.

Does that make sense?