Each charge or payment is booked against a corresponding invoice in FreeAgent. If an invoice for a given charge or payment already exists in FreeAgent, DoubleAgent uses it. If no such invoice exists DoubleAgent creates one.
When you create an invoice in FreeAgent you can have it emailed to your contact.
Now you can also have invoices created by DoubleAgent emailed in the same way to your contact. Invoices emailed like this show the payment received – which makes them useful for automatically sending invoices as (VAT) receipts. In fact I use this for sending out automatic payment receipts for DoubleAgent itself.