Custom Post Types relationships

From your question I gather your taxonomy term names match your category names. If this is the case and always will be the case, why not just query your custom post type by category slugs? Not sure what the exact name is, so I just made it video.

// get category slug
$cat = get_category( get_query_var( 'cat' ) );
$cat_slug = $cat->slug;

// query your custom post type (video?), assuming 
// slugs match for your category name and taxonomy term name
$my_args = array(
    'post_type' => 'video',
    'tax_query' => array(
        array(
            'taxonomy' => 'your_taxonomy_name',
            'field' => 'slug',
            'terms' => $cat_slug
        )
    )
);

$my_query = new WP_Query($my_args);

while ($my_query->have_posts()) {
    $my_query->the_post();
    // do the usual thing
}