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...