I had asked if the following is 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. Meaning, of course, that dinaturals are closed under composition iff *A* is a groupoid. Well, all I've received are requests for the proof. So: given *A* let P denote the covariant power-set functor on the category of sets. Fix an object C. Consider the bifunctor that sends A to P(C,A) and consider the dinatural transformation (A,A) -> P(C,A) that sends an endomorphism e in (A,A) to the set of solutions o the equation: x x e C -> A = C -> A -> A. The composition 1 -> (A,A) -> P(C,A) has as its unique value the _entire_ subset of maps from C to A. If it is dinatural then for any f:A -> C: P(C,A) / 1 | P(1,f) \ P(C,C). g f Every endomorphism of C is thus of the form C -> A -> C. In particular, the identity map is of that form, that is, f is left invertible, hence, every map targeted at C is left-invertible. If this remains true for every C then every map in *A* is left invertible, thus invertible. I also asked for a reference for: If *A* is a groupoid then *D* is equivalent to the category of presheaves on *A*. Nobody's asked for it, but let *A* be a groupoid anyway. For any target category and any dinatural S -> T one may easily prove that the hexagon used for defining dinaturals expands to a commutative diagram (from which the computability of dinaturals easily follows): SAA ------------------> TAA SfA / \ SAf TfA / \ TAf SBA SAB --> TBA TAB SBf \ / SfB TBf \ / TfB SBB -------------------> TBB. If *A* is a groupoid then the category composed of dinaturals between bifunctors from *A* to *B* is equivalent to the category composed of natural transformations between covariant functors from *A* to *B*. Show that for any bifunctor, S, there is an isomorphism (in the category composed of dinaturals) to a bifunctor, T, with the special property that TAB = TAA and TAx = TA1. Given S define T to be the bifunctor such that TAB = SAA and Tfg = Sff' (where f' is the inverse of f). The collection of identity functions from SAA to TAA forms a dinatural transformation as does the collection of identity functions from TAA back to SAA. The full subcategory of such functors is easily seen to be isomorphic to the category of contravariant functors from *A* to *B*.