This piece of code gets the categories used by a list of posts in Wordpress, this also can be filtered to only show categories that are children of an specific category

get_associated_terms.php

<?php

/**
 * This function will return a list of categories used by a group of posts ids
 *
 * @param string $taxonomy The name of the taxonomy to search for
 * @param array $objects_id The list of posts IDs to check against
 * @param integer $child_of A parent category to limit the search on
 *
 * @return array A list of terms
 *
 * @example getAssociatedTerms( 'product_cat', [ 16, 22, 32 ] )
 */
function getAssociatedTerms( $taxonomy = '', array $objects_id = [], $child_of = 0 ) {
    $args = [
        'taxonomy' => $taxonomy,
        'object_ids' => $objects_id,
        'child_of' => $child_of,
        'hide_empty' => true
    ];
    $query = new WP_Term_Query( $args );
    return $query->get_terms();
}