This might be a stupid mistake but I do not see where it is. Let $f:X \to Y$ be a morphism between smooth projective varieties (over $\mathbb C$), and let $F\in D^b(X)$, $G\in D^b(Y)$. Some version of Grothendieck-Verdier duality states that we have adjuctions $$f^* \dashv f_* \dashv f^!$$
The right hand side says $$\mathcal{Hom}(f_* F,G)={\mathcal Hom}(F, f^!G)$$ which is equivalent to $$(f_* F)^\vee \otimes G = F^\vee \otimes f^! G$$
The left hand side says $${\mathcal Hom}(f^*G,F)= {\mathcal Hom}(G, f_*F)$$ which is equivalent to $$(f^* G)^\vee \otimes F= G^\vee \otimes f_* F$$ Dualize this, we get $$f^* G \otimes F^\vee = G \otimes (f_* F)^\vee$$ Compare these two, we get $$f^* G= f^! G$$ which is definitely wrong, but where is the mistake? (Here everything like $f_*$, $f^*$, $\mathcal Hom$ is denoted as its derived functor)