Dinatural exercise

Is this new? Let *A* be a locally small category. Let *D* be the category whose objects are set-valued bifunctors on *A* (contravariant on the first variable, covariant on the second) and whose maps are the dinatural transformations. Then *A* is a groupoid. Well, OK, what I mean is that dinaturals are closed under composition iff *A* is a groupoid. (And, yes, this could all be done with the category of sets replaced with a topos.) I trust the following is old, but who has a reference? If *A* is a groupoid then *D* is equivalent to the category of presheaves on *A*.