wordpress function with an html dropdown to change an ajax request

function get_categories_dropdown( $cat_id ) {
    $args = array(
        'orderby' => 'name',
        'id' => $cat_id,
        'hide_empty' => false,
    );
    $categories = get_categories( $args );
    ?>
    <select id="cat" name="cat" class="postform">
        <?php
        foreach( $categories as $cat ):
            echo '<option value="'.$cat->term_id.'">'.$cat->name.'</option>';
        endforeach;
        ?>
    </select>
    <?php
}