Display category-specific comments in sidebar

Try this function:

function get_category_comments($category_id, $limit = 5) {

    global $wpdb;

    $sql = "
        SELECT {$wpdb->comments}.comment_ID
        FROM
            {$wpdb->comments},
            {$wpdb->posts},
            {$wpdb->term_taxonomy},
            {$wpdb->term_relationships}
        WHERE 1=1
            AND {$wpdb->comments}.comment_post_ID = {$wpdb->posts}.ID
            AND {$wpdb->term_relationships}.object_id = {$wpdb->posts}.ID
            AND {$wpdb->term_relationships}.term_taxonomy_id = {$wpdb->term_taxonomy}.term_taxonomy_id
            AND {$wpdb->comments}.comment_approved = '1'
            AND {$wpdb->term_taxonomy}.term_id = '{$category_id}'
        ORDER BY {$wpdb->comments}.comment_date DESC
        LIMIT 0, {$limit}
    ";
    $comments = array();
    foreach ($wpdb->get_results($sql) as $c)
        $comments[] = get_comment($c->comment_ID);
    return $comments;

}

Then you can have another function to display the comments:

function display_category_comments() {

    global $cat;
    if (!$category_id = intval($cat))
        return false;

    $comments = get_category_comments($category_id);
    ?>
    <ul>
    <?php foreach ($comments as $c) : $post = get_post($i = $c->comment_post_ID); ?>
        <li>
            <?php echo $c->comment_author; ?> - <a href="https://wordpress.stackexchange.com/questions/59462/<?php echo get_permalink($post->ID); ?>" title="Permalink to <?php echo $post->post_title; ?>"><?php echo $post->post_title; ?></a>
        </li>
    <?php endforeach; ?>
    </ul>
    <?php
}