A week or two ago, Neil Ghani asked about natural transformations between set-valued functors (I think they were set-valued, but anyway that is what my answer refers to and is probably true for any reasonably complete codomain category although a different argument would be required), say a: F ---> G, such that for any arrow f: A ---> B of the domain category, the square aA FA --------> GA | | | | |Ff |Gf | | | | v aB v FB --------> GB is a pullback. At the time, I sent Neil a private reply, but it bounced for some reason. (I said that that I thought that this condition was reasonable only when restricted to monic f and then such an a is called an elementary embedding.) Then a couple of people answered that it was called a cartesian arrow and I didn't try to resend my answer. Well, there is a simpler answer. In that generality, such an a is called a natural equivalence. In other words, non-trivial examples do not exist. To see this, it is useful to translate it, using Yoneda, into the following form. As usual, I will say that of two classes E and M of arrows in a category, E _|_ M (E is orthogonal to M) if in any diagram e A ----> B | | | | | | v m v C ----> D with e in E and m in M, there is a unique arrow B ---> C (called a diagonal fill-in) making both triangles commute. Let us denote by h^A the covariant functor represented by A and for f: A ---> B, denote by h^f, the induced natural transformation h^B ---> h^A. Let E be the class of all h^f. Then a is cartesian iff E _|_ {a}. Now suppose a is cartesian. First I show that a is monic (that is injective). If not, there is an object A and two different arrows u, v: h^A ---> F such that aA(u) = aA(v). Let E be the equalizer of u and v and let h^B ---> E be any arrow. Then the square B A h -----> h | | | |aA(u)=aA(v) | | v a v F -----> G has two diagonal fill-ins, u and v. Here the arrow h^B ---> h^A is the composite h^B ---> E ---> h^A and the arrow h^B ---> F is the composite h^B ---> E ---> h^A ---> F, the latter via u or v. In a similar way, we can show that a is surjective. In fact, given u: h^A ---> G, let E be a pullback of a and u and let h^B ---> E be arbitrary. Then we have a commutative square B A h -----> h | | | |u | | v a v F ------> G whose diagonal fill-in gives a lifting of u. It therefore seems appropriate to restrict the question to certain classes of arrows A ---> B, for example monics. Here are a couple of examples. If g: C --->> D is a regular (or just strict) epimorphism between objects of the domain category, then for E the class of h^f for all monic f, we have E _|_ {h^g}. Similarly if E is the class of h^f for all strict monic f and g is any epimorphism, E _|_ {h^g}.