<?php
// Call Wordpress Function get_categories ();
$args = array(
    'type'                     => 'post',
    'orderby'                  => 'name',
    'hide_empty'               => 0,
    'hierarchical'             => 1,
    'exclude'                  => '6, 1,16,17,18,19,20,15',
    'taxonomy'                 => 'category',
    'pad_counts'               => false
  );
$categories = get_categories($args);

foreach($categories as $category) {

// Input Construction
    // Grab each Category
    $catch_variable = $category->term_id;
    // Get the Category Description
    $description_content = category_description($catch_variable);
    // Limit amount of returned characters to break at the word level (not character)
    $limit_excerpt = substr($description_content, 0, strpos($description_content, ' ', 260));
    // Link to Category Archive, Excerpt I trimmed, & Second Link
    $cleancode = $limit_excerpt . ' &hellip;' . ' <a href="' . get_category_link( $category->term_id ) . '" ' . '>' . ' Topic Overview &rarr;'.'</a>';

// Construction of Output
    echo '<p class="topic-link-heading"><a href="' . get_category_link( $category->term_id ) . '" ' .
    'id="topic-link"'  . '>' . $category->name.'</a> </p>';
    $removephtml = substr($cleancode, 3);
    echo '<p class="topic-list">' . $removephtml . '</p>';

// Need Help Here
    $test = $category->term_id;
    echo $test;
    // Outputs the correct Category ID

// The Query
query_posts( array('category__and'=>$test));

// The Loop
while ( have_posts() ) : the_post();
    echo '<li>';
    the_title();
    echo '</li>';
endwhile;
$relevant_posts = query_posts( $post_args );
// Reset Query
wp_reset_query();
}

?>