Search products in a specific category

I am trying to extend the default behaviour of the search mechanism provided by the default module blocksearch in Prestashop 1.6.0.6.

I added to the default form a select input returning categories.

Everything is working in both controller and view sides. The whole form is sending the correct parameter to the SearchController. But, in Model, I still don't know what are the changes to apply on Search.php entity so that the method find returns products that belong to the specified category. I spent a good time trying to understand how the search mechanism in find works but to no avail. Can someone please point me the concerned part of code? where to add my conditions on category_id in SQL?

Edit: Among all the sql queries in Search::find method, I think the following one is the concerned. In fact, I added AND p.'id_manufacturer' ='.(int)$manufacturer_id.' and I can see changes. But for categories, it seems more complicated, I think I need a JOIN somewhere. I am afraid I build things that can apparently working but are created with no respect to best practices. Please to help me I am a newbie in sql.

$sql = 'SELECT p.*, product_shop.*, stock.out_of_stock, IFNULL(stock.quantity, 0) as quantity, 
                pl.`description_short`, pl.`available_now`, pl.`available_later`, pl.`link_rewrite`, pl.`name`,
             MAX(image_shop.`id_image`) id_image, il.`legend`, m.`name` manufacturer_name '.$score.', MAX(product_attribute_shop.`id_product_attribute`) id_product_attribute,
                DATEDIFF(
                    p.`date_add`,
                    DATE_SUB(
                        NOW(),
                        INTERVAL '.(Validate::isUnsignedInt(Configuration::get('PS_NB_DAYS_NEW_PRODUCT')) ? Configuration::get('PS_NB_DAYS_NEW_PRODUCT') : 20).' DAY
                    )
                ) > 0 new
                FROM '._DB_PREFIX_.'product p
                '.Shop::addSqlAssociation('product', 'p').'
                INNER JOIN `'._DB_PREFIX_.'product_lang` pl ON (
                    p.`id_product` = pl.`id_product`
                    AND pl.`id_lang` = '.(int)$id_lang.Shop::addSqlRestrictionOnLang('pl').'
                )
                LEFT JOIN `'._DB_PREFIX_.'product_attribute` pa ON (p.`id_product` = pa.`id_product`)
                '.Shop::addSqlAssociation('product_attribute', 'pa', false, 'product_attribute_shop.`default_on` = 1').'
                '.Product::sqlStock('p', 'product_attribute_shop', false, $context->shop).'
                LEFT JOIN `'._DB_PREFIX_.'manufacturer` m ON m.`id_manufacturer` = p.`id_manufacturer`
                LEFT JOIN `'._DB_PREFIX_.'image` i ON (i.`id_product` = p.`id_product`)'.
                Shop::addSqlAssociation('image', 'i', false, 'image_shop.cover=1').'
                LEFT JOIN `'._DB_PREFIX_.'image_lang` il ON (i.`id_image` = il.`id_image` AND il.`id_lang` = '.(int)$id_lang.')
                WHERE p.`id_product` '.$product_pool.'
                                AND p.`id_manufacturer` ='.(int)$manufacturer_id.'
                GROUP BY product_shop.id_product
                '.($order_by ? 'ORDER BY  '.$alias.$order_by : '').($order_way ? ' '.$order_way : '').'
                LIMIT '.(int)(($page_number - 1) * $page_size).','.(int)$page_size;

Edit numero 2 : temporary solution

if($category_id!=0)  $sql=$sql.' AND p.`id_product` IN (SELECT `id_product` FROM `ps_category_product` p WHERE p.`id_category`='.(int)$category_id.'  )';

You can try with this

$sql = 'SELECT p.*, product_shop.*, stock.out_of_stock, IFNULL(stock.quantity, 0) as quantity, 
            pl.`description_short`, pl.`available_now`, pl.`available_later`, pl.`link_rewrite`, pl.`name`,
         MAX(image_shop.`id_image`) id_image, il.`legend`, m.`name` manufacturer_name '.$score.', MAX(product_attribute_shop.`id_product_attribute`) id_product_attribute,
            DATEDIFF(
                p.`date_add`,
                DATE_SUB(
                    NOW(),
                    INTERVAL '.(Validate::isUnsignedInt(Configuration::get('PS_NB_DAYS_NEW_PRODUCT')) ? Configuration::get('PS_NB_DAYS_NEW_PRODUCT') : 20).' DAY
                )
            ) > 0 new
            FROM  `'._DB_PREFIX_.'category_product` cp
            LEFT JOIN `'._DB_PREFIX_.'product` p
                ON p.`id_product` = cp.`id_product`

            '.Shop::addSqlAssociation('product', 'p').'
            INNER JOIN `'._DB_PREFIX_.'product_lang` pl ON (
                p.`id_product` = pl.`id_product`
                AND pl.`id_lang` = '.(int)$id_lang.Shop::addSqlRestrictionOnLang('pl').'
            )

            LEFT JOIN `'._DB_PREFIX_.'product_attribute` pa ON (p.`id_product` = pa.`id_product`)
            '.Shop::addSqlAssociation('product_attribute', 'pa', false, 'product_attribute_shop.`default_on` = 1').'
            '.Product::sqlStock('p', 'product_attribute_shop', false, $context->shop).'
            LEFT JOIN `'._DB_PREFIX_.'manufacturer` m ON m.`id_manufacturer` = p.`id_manufacturer`
            LEFT JOIN `'._DB_PREFIX_.'image` i ON (i.`id_product` = p.`id_product`)'.
            Shop::addSqlAssociation('image', 'i', false, 'image_shop.cover=1').'
            LEFT JOIN `'._DB_PREFIX_.'image_lang` il ON (i.`id_image` = il.`id_image` AND il.`id_lang` = '.(int)$id_lang.')
            WHERE p.`id_product` '.$product_pool.'
                            AND p.`id_manufacturer` ='.(int)$manufacturer_id.' AND cp.id_lang ='.(int)$id_category.'
            GROUP BY product_shop.id_product
            '.($order_by ? 'ORDER BY  '.$alias.$order_by : '').($order_way ? ' '.$order_way : '').'
            LIMIT '.(int)(($page_number - 1) * $page_size).','.(int)$page_size;

where (int)$id_category is the id_category (from the select) and category_product is the table storing the relation between products and categories

Hope this helps.