this may a good deeper read on the subject [Yoneda theory for double categories - R Paré - Theory and Applications of Categories, 2011 ](http://www.tac.mta.ca/t

this may a good deeper read on the subject [Yoneda theory for double categories - R Paré - Theory and Applications of Categories, 2011 ](http://www.tac.mta.ca/tac/volumes/25/17/25-17.pdf http://www.tac.mta.ca/tac/volumes/25/17/25-17.pdf)