Cameron,
That code is going to go right in here:
Code:
{section name=cat_num1 loop=$categories_level_2}
{if $categories_level_2[cat_num][0][cat_num1].categoryid ne ""}<option value="{ $categories_level_2[cat_num][0][cat_num1].categoryid}" {if $smarty.get.in_category eq $categories_level_2[cat_num][0][cat_num1].categoryid or $cat eq $categories_level_2[cat_num][0][cat_num1].categoryid}selected{/if}>
{$categories_level_2[cat_num][0][cat_num1].category|escape}
</option>{/if}
{/section}
Replace the
{$categories_level_2[cat_num][0][cat_num1].category|escape} with
{$categories_level_2[cat_num][0][cat_num1].category_name|escape}.
Hope that helps.