Suppose $X,Y$ are smooth varieties over $\mathbb{C}$, and let $K\_i \in D^b(X \times Y), i=1,2$ be objects in the derived category of bounded complex of coherent sheaves on $X \times Y$. Then there are Fourier-Mukai functors (FM functors) $\Phi\_{K\_i}$ from $D^{b}(X)$ to $D^b(Y)$ associated to kernel $K\_i$:


$$F \mapsto q\_{\*}(K\_i\otimes p^\*(F)).$$ (here pullback, tensor and pusshfoward are all in the derived sense, and $p: X\times Y \to X$, $q: X \times Y \to Y$ are projections.)


My question is: if there is a natural transformation $G: \Phi\_{K\_1} \to \Phi\_{K\_2}$ respect derived categories, then does $G$ necessary come from a morphism of kernel $K\_1 \to K\_2$?


Somehow, I feel if one can write $\Phi\_{K\_1} \circ \Phi^{-1}\_{K\_2}$ (I use $\Phi^{-1}\_{K\_2}$ to denote some adjunct functor of $\Phi\_{K\_2}$), then it is also a FM transform, and by the uniqueness of FM kernel, one should get some morphism between the kernels. But I am not quite sure...



Morphism between Fourier-Mukai functors implies the morphism between kernels?
This is somewhat sad, but I think (part of) what we've learned from the whole triangulated-vs-dg story is the following pseudo-statement: the bare category of functors Fun(D(X),D(Y)) is the wrong thing to take -- it should be replaced the category D(X x Y).


There is a nice example, due to the usual Bondal-Orlov-Bridgeland people ([Example 6.5 in Caldararu's notes](http://arxiv.org/pdf/math/0501094v1.pdf)), which illustrates this perfectly.


Thus, if you find something weird happening in Fun(D(X),D(Y)) you shouldn't be discouraged -- it's not your fault.


Of course, there are big names in the field (Canonaco and Stellari to name two) who might disagree with my initial pseudo-statement. It really depends on what you care about.


