Thank you for you reply, I play a federation captain and the DOFF I bought have the federation symbol on the upper right corner of the picture and the color frame the picture is of Federation blue. I don't think I bought the wrong faction, did I?
The trouble is that the exchange is a bit buggy, putting it nicely, and can show DOFFs as Fed when they aren't. I have fallen victim to this myself a couple of times when I first started buying DOFFs off the exchange. You can use the option to show only usable to you on the exchange as this will remove those you can't use.
In your DOFF roster does it still show the fed colour?
Also the actual DOFF interface can bug out at times. I have chosen a DOFF assignment to do and it shows no recommendations even though I know for a fact I have the relevant DOFFS for it. If this happens you click on the DOFF slot under the assignment description and the recommendations appear.