Yes it is just a CofA Annual, but that statement hides a much deeper potential problem.
One can fit all sorts of stuff to a Private CofA plane which could not be fitted to a Transport CofA plane.
Such items would need to either be certified (I've known of certain JAR145 firms that would generate the required piece of paper for a 3 figure fee; whether this is legal I have no idea but it shows what a scam the certification business is; the actual part involved would be exactly identical except that it didn't have a JAR1 form when originally fitted) or would have to be removed.
One gets similar issues from going G to N, or N to G.
If a plane was previously on a Transport CofA (nearly all CofA planes do come with a Transport CofA when brand new) and nothing was changed since then all that should be needed is for the maintenance to be brought up to date, and the paperwork. Perhaps a grand or two, including the £500-700 CAA CofA fee.