I was trying to understand the Mukai Fourier transform from his paper : Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math Journal, 1981.

I am not very familiar with Derived categories and even less familiar with $D^-(X)$ and so this question, as Mukai works with $D^-(X)$. My knowledge of derived categories has been obtained from Iversen's book Cohomology of Sheaves.

I usually think of the bounded below derived category $D^+(X)$ by an analogy with injective resolutions modulo quasi isomorphism/homotopy equivalence. For example, $f:X\to Y$ is a morphism of algebraic varieties, then we can define the higher direct images $R^if_*F$ for any sheaf $F$ on $X$. If we have a short exact sequence $0\to F\to G\to H\to 0$, then we can find injective resolutions which are term wise split and we can get long exact sequences of higher direct images. Now instead of thinking of each of the $R^if_*F$ individually we could just take the whole complex $f_*I^\bullet$ and call this $Rf_*I^\bullet$, where $0\to F\to I\bullet$ is an injective resolution.

Since I have mostly seen only injective resolutions in action, could someone explain to me the motivation for working with $D^-(X)$, where I guess one would have to work with projective resolutions, which may not always exist.

For my purpose, I convinced myself about Proposition 1.3 in the paper of Mukai by working with $D^+(X)$ as in the case of interest (abelian varieties) the projection morphisms are flat and we are tensoring with a flat sheaf.

Thanks in advance.