continues(loop) page navigation with get_permalink()

Hello,
I already searched the forum with this problem and I found a lot of good information but I’m still not possible to solve my problem. I’m sorry about that.

I would like to turn my page navigation in to a loop. I’m quit close but my problem is that I’m not able to switch between the categories. At the moment I have a loop inside of the categorie but I’m not able to jump to the next categorie.

My script:

<?php
$prev_post = get_adjacent_post( true, '', true );
$next_post = get_adjacent_post( true, '', false );
$prev_post_id = $prev_post->ID;
$next_post_id = $next_post->ID;
?>

<?php
if(($prev_post_id === '') || ($prev_post_id === null)) :

$query = new WP_Query(array('order' => 'DESC', 'posts_per_page' => '1', 'cat=4'));

if($query->have_posts()) : while($query->have_posts()) :
    $query->the_post();
    $prev_post_id = get_the_ID();
    endwhile;
endif;

wp_reset_postdata();

endif;

if(($next_post_id === '') || ($next_post_id === null) ) :

$query = new WP_Query(array('order' => 'ASC', 'posts_per_page' => '1', 'cat=4'));

if($query->have_posts()) : while($query->have_posts()) :
        $query->the_post();
        $next_post_id = get_the_ID();
    endwhile;
endif;

wp_reset_postdata();

endif;
?>

<div id="left">
<a href="<?php echo get_permalink($prev_post_id); ?>" class="myCursorPrev">&nbsp;</a>
</div>
<div id="right">
<a href="<?php echo get_permalink($next_post_id); ?>" class="myCursorNext">&nbsp;</a>
</div>

$query = new WP_Query(array('order' => 'ASC', 'posts_per_page' => '1', 'cat=4'));
I tried different usage with the cat=4 or cat=1,2,3,4 but it has no effect.

thanks for any help.
best, carol