Because of the symmetry inherent in the Cartesian monoidal structure of the category of sets, there is no difficulty in treating on an equal footing a category and its opposite. Consequently, once one understands what to mean by a (covariant) functor from one category to another, contravariant functors can be explained with the greatest of ease as either covariant functors from the first category to the opposite of the second, or, equally effectively, as covariant functors from the opposite of the first category to the second itself. The only small snag here is that it is the orientation of the maps in the target category that determines the natural direction of the maps from one functor to another in the functor category. But this problem already rears its head in the identification of covariant functors fromAtoBwith covariant functors fromto A^{op}and is dealt with simply by making a well-motivated choice. B^{op},

Symmetry comes into play again in defining the product of two (or more) categories, and, hence, in explaining what to mean by functors of two (or more) variables, of fixed, or mixed, variance: one takes them, for example, to be (covariant) functors from a productto A^{op}×BC.

A much larger and less easily surmountable snag arises once one wants to generalize the observations above to the setting of categories enriched in a closed, or monoidal, or, most generally, multilinear categoryVwhose structure (like that of almost any endofunctor categoryunder the monoidal operation of composition) manifests no inherent symmetry whatsoever. Here, given a A^{A}V-categoryA, there may well benoappropriate understanding at all for what theopposite of V-categoryAshould be. Hand in hand with this difficulty comes the further annoyance that, despite the easy availability of a notion of covariantfrom one V-functorto another, absence of symmetry totally stymies any attempt to elucidate what a V-categorycontravariantbetween two V-functorshould be. By the same token, absence of symmetry makes it impossible to explain what to mean by a V-categoriesof several variables, of fixed, or mixed, variance. V-functor

On the other hand, even when the multilinear base categoryVis not itself a(i.e., is not closed) – even if, say, V-categoryVis non-symmetrically monoidal, like ourexample earlier – it miraculously remains possible to explain what to understand by A^{A}V-valued– of V-functorseithervariance – defined on an arbitraryV-categoryA. What is more, even a notion ofV-valuedof mixed variance, defined on a pair of V-functorV-categoriesAandB, contravariant inAand covariant inB, is readily available: this comes down to a systemFconsisting of a rule assigning to each objectAofAand each objectBofBa value-objectof F(A,B)V, along with further information providingmaps V-multilinearall satisfying conditions analogous to the usual associativity and unit laws. Such anF: <B(B,B'),F(A,B),A(A',A) >F(A',B')Fwould be termed aV-valuedof the two variables V-functorAandB, contravariant inAand covariant inB. A prime example of such a functor is the actualV-valuedhom-functor of theV-categoryAitself.

What is more, despite the likely absence of anyon which such a V-categoryV-valuedof two variables might be defined, there is available a “law of exponents,” of sorts, identifying V-functorV-valuedof mixed variance, contravariant in V-functors,Aand covariant inB, with actual (covariant)on V-functorsB– taking values (whenAis small enough, andVcomplete enough) in a sort of presheafV-categoryof contravariant V^{(Aop)}V-valuedon V-functorsA. Applied toown A’shom-functor, this identification yields a Yoneda representation V-valuedwhich (sure enough) turns out to be AV^{(Aop)},and-faithful. Even more surprisingly, there is a twist on this “law of exponents” permitting identification of such V-full-V-valuedof mixed variance with, instead, single covariant V-functorsfrom V-functorsAto abest described, intuitively, as the V-categoryV-opposite( of the category of covariantV^{B})^{op}V-valuedon V-functorsB.Note: NOT defined on– there may well not BE any A^{op}serving as opposite for V-categoryA; and NOT taking values in– there may well not be even a reasonable candidate (no matter how small V^{B}Bor how completeVmay be) for a “covariantV-valuedon V-functorsB”V-category; BUT defined on V^{B}Aand taking values in( This identification, applied to theV^{B})^{op}.hom functor of V-valuedyields the “other” Yoneda representation, A,which, like the former one, is again A(V^{A})^{op},and-faithful. V-full-

An amusing corollary of these considerations and others like them is the following moral regarding the direction of morphisms in Kleisli categories, and the variance of Lawvere-style algebras as (roughly speaking) “representable” functors on such Kleisli categories. One way to concoct the Kleisli category of apair V-adjoint(with F|–U:XAleft adjoint to F:AXis to form the U:XA)full imageofF, that is, thewith same objects as V-categoryAbutof-morphisms V-objects-Another way, even in the conceivable absence of [A,B]=X(F(A),F(B)).F, is to use as morphisms fromAtoBa suitableof V-objecttransformations between V-naturaland U^{A}(here U^{B}is the U^{A}:XVgiven by V-functorNow while it has traditionally been thought that it should be the transformations from U^{A}(X) =A(A,U(X)) ).to U^{A}– mimicking the direction of U^{B}operations – that ought to serve as the Kleisli-maps from A-aryAtoB, we see here that, it being notbut V^{X}( that stands any chance of being aV^{X})^{op}we really need to use V-category,( whose “elements” are theV^{X})^{op}(U^{A},U^{B}) ,transformations from V-naturalto U^{B}(!), as the Kleisli maps from U^{A}AtoB. Fortunately, this observation is consistent with (rather than opposite to) the one provided by the full image ofF, for, by Yoneda and adjointness,[A,B]=

X(F(A),F(B)) ≈ (V^{X})^{op}(Y(F A),Y(F B)) ≈ (V^{X})^{op}(U^{A},U^{B}) .

And theof Lawvere-style algebras “over” the Kleisli V-categoryV-categoryKcan then be taken as the fullof the V-subcategoryV-categoryof contravariant V^{(Kop)}V-valuedon V-functorsKwhose compositions withare explicitly representable (which is to say: as the pullback of the diagram AK

V^{(Kop)}↓ AV^{(Aop)}

ofand V-categoriesV-functors).

Historical remarks.Much of what is to be told here has appeared in the author’s ancient articles in Springer LNM ## 99 and 195, or was promulgated orally on the occasion of the esteemed Professor Charles Ehresmann's 70^{th}birthday celebration at Chantilly/Fontainebleau in the summer of 1975. But the oral seeds seem never to have taken root, having instead been simply scattered by the winds of time, so it has seemed worthwhile to broadcast them once again, hoping that this time they will fall on more fertile soil.

Technical remarks.It is of course a gross abuse of language to speak of either of these functor categoriesV^{(Aop)},( asV^{A})^{op}The actual structure available is, at best, for each pair of V-categories.V-functorsF,G, a “job-description” for the desiredhom-object V-ishin the form of a suitable (n.t.F,G) ,functor on Sets-valued– or better, on V^{op}( the (opposite of the) strictly associative monoidal categoryM_{0}(V))^{op},(of finite strings of objects of M_{0}(V)V) that serves (in SLNM 195) as themultilinear structureofV– withSetsany available category of “sufficiently large” sets. Any object ofVrepresenting this “job-description” functor will serve as the desiredtransformations object. V-natural

27 May 2005, New Haven, CT (USA). All rights reserved.