Displaying list of subcategories and not parent category

Here’s a function that get’s the category from the post and then displays an unordered list of term children links. I’m assuming you only have one category per post. If you don’t, then you’d have to iterate through $cat_id in addition to $cat_children:

// get category id
$cat_id = get_the_category();
// get array of all children of the first category assigned to the post
$cat_children = get_term_children( $cat_id[0]->term_id, 'category' );
// start building our list
$cat_children_list="<ul>";
foreach( $cat_children as $child ) {
    // get the term object of each child term
    $term = get_term( $child, 'category' );
    // output the list item with link and label
    $cat_children_list .= '<li><a href="' . get_term_link( $term ) . '">' . $term->name . '</a><?li>';
}
// finish up the list
$cat_children_list .= '</ul>';
// echo the output
echo $cat_children_list;