Does surrogate key assignment for a fact table require that the source data has natural keys?

Question

Let's say I have a simple OLTP database with orders, products, and customers:

And from it, I am building a data mart with an orders fact table, product dimension, customer dimension, and date dimension:

When loading the orders table into fact_orders (let's say I was using an SSIS Lookup Transformation to assign the surrogate keys), does that mean that the source of data for the orders would also need to have the natural "foreign key" values that were associated with the order in the OLTP system?

In other words, would the data that's being loaded come from a query like this?

SELECT 
  order_date,                   -- needed to get date surrogate key
  customer_name,                -- needed to get customer surrogate key
  product_name,                 -- needed to get product surrogate key
  order_number,                 -- denegenerate dimension,
  qty_ordered AS order_qty,     -- measure
  total_amount AS order_amount  -- measure
FROM orders o 
  INNER JOIN customers c 
    ON o.customer_id = c.customer_id
  INNER JOIN products p 
    ON o.product_id = p.product_id