Assumptions:
You don’t explain how you want to install the Font Awesome package, so I just assume for the moment that you use the plugin Font Awesome Icons.
You wrote:
Before anyone says use a background image, I do not want to do that. I
would like it to be physical.
so I assume you want to use the <i> tag directly, for example:
<i class="fa icon-caret-right"></i>
after each category link in the widget category list.
Idea:
You can use the wp_list_categories filter to modify the output of the widget category list.
Example:
Here is a simple example how to inject it into the category list via the wp_list_categories filter:
/**
* Inject Font Awesome <i> tag after each widget category link
*
* @param string $output
* @return string $output
*/
function custom_wp_list_categories( $output )
{
remove_filter( current_filter(), __FUNCTION__ );
return str_ireplace( '</li>', '<i class="fa icon-caret-right"></i></li>', $output);
}
add_action( 'widgets_init', function(){
add_filter( 'wp_list_categories', 'custom_wp_list_categories' );
});
This will give you an output similar to this one:
