AFAIK yes, he is. IIRC he writes the captains notes for the matchday programmes anyway.
As much as I'd like to see Adam get it there's 2 reasons I wouldn't give it too him yet
1)He's going to have a huge amount of pressure on him to perform & he could do without the extra that captaincy brings.
2)I don't think that wingers\attacking midfielders make the best captains I prefer defensive minded players or central midfielders as they tend to have better views of the whole pitch.
Personally I'd give it to Jos, or maybe Morgan. If KD is no longer first choice keeper then Adam should take over as club captain IMO.