WordPress query two categories at once - ex. http://website.com/?category_name=cats+dogs

gistfile1.php

<?php 
function duffy_category_search_logic( $query ) {

	if ( !isset( $query->query_vars[ 'category_name' ] ) )
		return $query;
		
	// split cat query on a space to get IDs separated by '+' in URL
	$cats = explode( ' ', $query->query_vars[ 'category_name' ] );
		
	if ( count( $cats ) > 1 ) {
		unset( $query->query_vars[ 'category_name' ] );
		$query->query_vars[ 'category__and' ] = $cats;
	}
	
	return $query;
}
add_action( 'parse_request', 'duffy_category_search_logic', 11 );

gistfile2.phtml

<form role="search" class="searchform custom" method="get" id="searchform" action="<?php echo home_url( '/' ); ?>">
  <input type="hidden" name="category_name" value="cats dogs" />
  <input type="text" name="s" id="s" class="s" placeholder="Search" />
  <input class="button" type="submit" id="searchsubmit" value="Submit" />
</form>