Set cookies cross-domain with Nextjs13 api route

Question

I'm having my Nextjs 13 app and another app which is a shop. Now i want to click on a product from my nextjs app which would send the cookies to my shop app and thus update the cart.

I am aware that i have to allow CORS and set "Access-Control-Allow-Origin", "your-origin.com" and "Access-Control-Allow-Credentials", "true" in my shop app and i did that.

This is the api code to access my shop and update the cart:

const responsee = await fetch(
      "https://example.com/index.php?/cart/v1",
      {
        method: "POST",
        headers: {
          Accept: "application/json, text/javascript, */*; q=0.01",
          "Accept-Language": "de,en-US;q=0.7,en;q=0.3",
          "Accept-Encoding": "gzip, deflate, br",
          "X-CSRF-Token": "123",
          "Content-Type": "application/x-www-form-urlencoded; charset=UTF-8",
          "X-Requested-With": "XMLHttpRequest",
          Origin: "https://example.com/",
          DNT: "1",
          Connection: "keep-alive",
          Referer: "https://example.com/index.php?/products/",
          Cookie: "SESSIDe08a=123abc; Cart-Session=123abc",
          "Sec-Fetch-Dest": "empty",
          "Sec-Fetch-Mode": "cors",
          "Sec-Fetch-Site": "same-origin",
          Pragma: "no-cache",
          "Cache-Control": "no-cache",
        },
        body: "cat_id=4&domain[id]=13&domain[name]=NEWDOMAINNAME.at&domain[period]1&domain[type]=register",
      }
    );

Now, pardon my ignorance (my backend knowledge is very limited), how do i set cookies to mydomain.com(my shop app) from my nextjs api route assuming i'm allowing CORS for that?

I'd appreciate any help or a link with an example.