Agree with posts above. Black smoke, if that's really what it is, is typically indicative of a seriously over-rich mixture. The ECU will always try to manage fuel trim to prevent this, and if it cannot, the forward O2 sensors will be driving the ECU to the far limits of the fuel trim tables and a code will appear as a result.
During cold start, the engine is allowed (in fact, is expected) to run a bit rich and a bit high in RPM, but is not normally enough to cause 'black smoke' as a result, even if the catalyst isn't up to temperature enough to burn off any remaining hydrocarbons in the exhaust.
Like others, I'd be interested in a video if you can get someone to take one while you perform a cold start and wait whatever time until the 'black smoke' subsides. How long is this time, typically?