display category name based on its equivalent id

to convert it from category id to category name

when I echo $slidecat it shows cat id of “burger” which is 16

You can use get_term_field() to get the term/category slug by ID:

// Get the slug and then pass it to WP_Query.
$cat_slug = get_term_field( 'slug', $slidecat, 'product_cat' );

$args = array(
    'post_type'      => 'product',
    'posts_per_page' => 3,
    'product_cat'    => $cat_slug,
    'orderby'        => 'date',
    'order'          => 'ASC'
);

Or you could actually just pass the category ID to WP_Query using the taxonomy query, like so:

$args = array(
    'post_type'      => 'product',
    'posts_per_page' => 3,
//  'product_cat'    => 'burger', // don't set product_cat
    'tax_query'      => array(
        array(
            'taxonomy' => 'product_cat',
            'terms'    => $slidecat,
            'field'    => 'term_id',
        ),
    ),
    'orderby'        => 'date',
    'order'          => 'ASC'
);

Additionally, you should replace the wp_reset_query() in your code with wp_reset_postdata() because the former is used only if the main WordPress query is modified (e.g. if you called query_posts() which should just be avoided).

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