Let $M$ be a complex and $K$ be a module which is viewed as a complex concentrated in degree 0. I'm wondering what sign should a canonical map $\Phi\colon\text{Hom}(M, K)[a-b]\to \text{Hom}(M[b], K[a])$ impose on $f\in\text{Hom}(M^n, K)$.
As shown at the bottom of Sign rules -- The Stacks Project, the sign $(-1)^{nb}$ is a natural choice.
However, if we decompose $\Phi$ into two maps $\text{Hom}(M, K)[a-b]\to \text{Hom}(M, K[a])[-b]\to \text{Hom}(M[b], K[a])$, the same reasoning shows that the first map gives no sign and the second one gives a sign of $(-1)^{(n+a)b}$.
If I haven't got anything wrong, there seems to be a difference of signs when calculating in different ways. Then which one would be the correct sign?
