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We will make a view, load the term from contextual argument, get its weight into $static variable, and check on each row if row's term's weight is higher than our arg's one.


Create a view of terms, order them by weight.
Display only 1 item.
Add and hide taxonomy term: weight field.
Add a contextual filter : Taxonomy term: Term ID
Add a PhP filter and :


Setup code :

$static = taxonomy_term_load($view-&gt;args[0]);
$static = intval($static-&gt;weight);


Filter code :

if(intval($data-&gt;taxonomy_term_data_weight) &lt; $static) {return TRUE;}


If you're sure that your terms have following weight, you can use that filter code : 

if(intval($data-&gt;taxonomy_term_data_weight) == intval($static - 1) || intval($data-&gt;taxonomy_term_data_weight) == intval($static + 1) ) {return TRUE;}


to get the next and following terms. But careful with 0 and last values... :)

And of course, in that case, you have to display all items in the view.