<?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
    $category = $category->term_id;
    echo $category;
    // Outputs the correct Category ID

 $post_args = array(
'posts_per_page'   => 5,
'offset'           => 0,
'category'         => '$category',
'orderby'          => 'post_date',
'order'            => 'DESC',
'include'          => '',
'exclude'          => '',
'meta_key'         => '',
'meta_value'       => '',
'post_type'        => 'post',
'post_mime_type'   => '',
'post_parent'      => '',
'post_status'      => 'publish',
'suppress_filters' => true );

// The Query
query_posts( $post_args );

// The Loop
while ( have_posts() ) : the_post();
    echo '<li>';
    the_title();
    echo '</li>';
endwhile;

// Reset Query
wp_reset_query();


}
?>