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I want to say that the above answer worked for me, but I needed to make some slight adjustments to the if statement. I also made an adjustment to work like...

if the post category is a parent, display the children

if the post category is a parent but has no children, display posts

<?php 

$cat_id = get_query_var('cat');

if( ! empty( $cat_id ) ) {
    $category = get_category( $cat_id, ARRAY_A );

    if( ! empty( $category ) ) {
        // parent category
        if( $category['parent'] === '0' ) {

            // get child IDs
            $terms = get_term_children( $cat_id, 'category' );

            //checks to see if children exist
            if ($terms[0] != '') {

                foreach( $terms as $term_id ) {
                    $child_category = get_term( $term_id, 'category' );

                    // if you need the category
                    $category_name = $child_category->name;


                    /**
                     * Sample data in your child category object: 
                     * 
                     * object(stdClass)[365]
                              public 'term_id' => string '14' (length=2)
                              public 'name' => string 'Child 1' (length=7)
                              public 'slug' => string 'child-1' (length=7)
                              public 'term_group' => string '0' (length=1)
                              public 'term_taxonomy_id' => string '14' (length=2)
                              public 'taxonomy' => string 'category' (length=8)
                              public 'description' => string '' (length=0)
                              public 'parent' => string '12' (length=2)
                              public 'count' => string '5' (length=1)
                              public 'object_id' => string '33' (length=2)
                     * 
                     * 
                     */

                    // do whatever you like with the categories now...
                }
            }  
            else { // in child category
                // do the regular loop here, if ( have_posts() ) ...
            }
        }
    }
}