我知道了...必须运行单独的SELECT函数：一个用于获取类别列表，然后在该循​​环内提供另外一个函数以计算该类别中有多少个条目。我本来希望将这两个循环合而为一，但这对我来说很有效。

I figured it out... had to run to separate SELECT functions: one to fetch the list of categories, and then one more functions within that loop to count how many entries there is within the category. I would have preferred to have these two loops as one, but this works out for me.

<?php

// This will get us a list of the categories that our Author has published in
$author = get_query_var('author');
$categories = $wpdb->get_results("

SELECT DISTINCT(terms.term_id) as ID, terms.name, terms.slug, tax.description
FROM $wpdb->posts as posts
LEFT JOIN $wpdb->term_relationships as relationships ON posts.ID = relationships.object_ID
LEFT JOIN $wpdb->term_taxonomy as tax ON relationships.term_taxonomy_id = tax.term_taxonomy_id
LEFT JOIN $wpdb->terms as terms ON tax.term_id = terms.term_id
WHERE posts.post_status = 'publish' AND
    posts.post_author = '$author' AND
    tax.taxonomy = 'category' 
ORDER BY terms.name ASC
");


// This loop picks up categories
foreach($categories as $category) : 

$catid = $category->ID;

// Now, inside the loop, we need to count how many posts that the Author has published.
$counter = "SELECT COUNT(*)
FROM $wpdb->posts
LEFT JOIN $wpdb->term_relationships ON($wpdb->posts.ID = $wpdb->term_relationships.object_id)
LEFT JOIN $wpdb->term_taxonomy ON($wpdb->term_relationships.term_taxonomy_id = $wpdb->term_taxonomy.term_taxonomy_id)
WHERE $wpdb->term_taxonomy.term_id = $catid
AND $wpdb->term_taxonomy.taxonomy = 'category'
AND $wpdb->posts.post_status = 'publish'
AND post_author = '$author'
";

$user_count = $wpdb->get_var($counter);

echo '<div class="archive_author">' . $category->name . '<br/><span class="subcounter">' . $user_count . ' posts</span></div>';

endforeach; 

?>