We must include event handlers before sending a request to detect when the operation is complete. Your request passes through several states before the response is returned. The onreadystatechange property detects when the request is complete. When the readystate property equals 4, the request is complete. You can check the status property at this time.

The response of our request in JSON format is as follows:

Therefore, here is the complete code of the GET request that we need to send:

function retrieveProductInfo()
{
  //Gets attribute for product.
  var product = Xrm.Page.getAttribute("dyn_product");
  //Attribute name of Category field
  var categoryField = "dyn_category";
  //Attribute name of Sub Category Field.
  var subCategoryField = "dyn_subcategory";
  //Variable to store product ID
  var productId;
  //Check if Product attribute is null
  if (product != null)
  {
    //Assign the selected product Id
    productId = product.getValue()[0].id.replace(/[{}]/g, "");
  }
  //Initializes request object
  var req = new XMLHttpRequest();
  //Opens the request
  req.open("GET", Xrm.Page.context.getClientUrl() + 
  "/api/data/v8.2/dyn_customproducts(" + productId + ")?
  $select=_dyn_category_value,dyn_name,_dyn_subcategory_value", true);
    //Set the request headers
    req.setRequestHeader("OData-MaxVersion", "4.0");
    req.setRequestHeader("OData-Version", "4.0");
    req.setRequestHeader("Accept", "application/json");
    req.setRequestHeader("Content-Type", "application/json; 
    charset=utf-8");
    req.setRequestHeader("Prefer", "odata.include-annotations="*"");
    //Function to detect changes of readystate
    req.onreadystatechange = function ()
    {
      if (this.readyState === 4)
      {
        req.onreadystatechange = null;
        //Checks if request was completed succesfully
        if (this.status === 200)
        {
          //Parses the JSON response
          var result = JSON.parse(this.response);
          //Get the CategoryId
          var categoryId = result["_dyn_category_value"];
          //Gets Category Name
          var categoryName = result["_dyn_category_value@OData.
          Community.Display.V1.FormattedValue"];
          //Gets logical name of Category entity.
          var categoryLogicalName = 
          result["_dyn_category_value@Microsoft.
          Dynamics.CRM.lookuplogicalname"];
          //Get the SubCategoryId
          var subcategoryId = result["_dyn_subcategory_value"];
          //Get the SubCategory Name
          var subcategoryName = result["_dyn_subcategory_value@OData.
          Community.Display.V1.FormattedValue"];
          //Get the logical name of SubCategory Entity
          var subcategoryLogicalName = 
          result["_dyn_subcategory_value@Microsoft.
          Dynamics.CRM.lookuplogicalname"];
          //Calls the funtion to set lookups
          SetLookupValue(categoryField, categoryId, 
          categoryName, categoryLogicalName);
          SetLookupValue(subCategoryField, subcategoryId, 
          subcategoryName, subcategoryLogicalName);
          }
          else
          {
            Xrm.Utility.alertDialog(this.statusText);
          }
        }
      };
      req.send();
    }
    //Function that sets lookups

    function SetLookupValue(fieldName, id, name, entityType)
    {
      if (fieldName != null)
      {
        var lookupValue = new Array();
        lookupValue[0] = new Object();
        lookupValue[0].id = id;
        lookupValue[0].name = name;
        lookupValue[0].entityType = entityType;
        if (lookupValue[0].id != null)
        {
          Xrm.Page.getAttribute(fieldName).setValue(lookupValue);
        }
     }
  } 

So, by performing the preceding steps, the linked entities that were used to set the other two Lookup fields on the form can be retrieved through the Web API.