} elseif ( 'term_taxonomy_id' == $field ) {
    $value = (int) $value;
    $_field = 'tt.term_taxonomy_id';

    // No `taxonomy` clause when searching by 'term_taxonomy_id'.
    $tax_clause = '';
} else {
    $term = get_term( (int) $value, $taxonomy, $output, $filter );
    if ( is_wp_error( $term ) || is_null( $term ) ) {
        $term = false;
    }
    return $term;
}

$term = $wpdb->get_row( $wpdb->prepare( "SELECT t.*, tt.* FROM $wpdb->terms AS t INNER JOIN $wpdb->term_taxonomy AS tt ON t.term_id = tt.term_id WHERE $_field = %s $tax_clause LIMIT 1", $value ) );
if ( ! $term )
    return false;

// In the case of 'term_taxonomy_id', override the provided `$taxonomy` with whatever we find in the db.
if ( 'term_taxonomy_id' === $field ) {
    $taxonomy = $term->taxonomy;
}