Hi there,
thats interesting. Something we can consider.
You can do this with a Hook Element.
This is the PHP required to display the first category:
<?php
$terms = get_the_terms( get_the_ID(), 'product_cat' );
if ( $terms && ! is_wp_error( $terms ) ) :
if ( ! empty( $terms ) ) {
echo $terms[0]->name;
}?>
<?php endif;?>
The Hook would be the:
woocommerce_after_shop_loop_item_title
I think the review stars are hooked in at 5 so the priority would need to be 4 or lower.