Background
Dimca's book "Sheaves in topology" Theorem 2.3.29 (Projection Formula). Let $f : X \rightarrow Y $ be a continuous map, $ \mathcal{F^{\cdot}} \in D^{-}(X), \mathcal{G^{\cdot}} \in D^{-}(Y)$. Then there is a natural isomorphism in $D^{-}(Y),$ $Rf_!\mathcal{F^{\cdot}} \otimes^L \mathcal{G^{\cdot}} \cong Rf_! (\mathcal{F^{\cdot}} \otimes^L \mathcal{G^{\cdot}})$.
In order to define $Rf_!\mathcal{F^{\cdot}}$, we need to replace $\mathcal{F^{\cdot}}$ by an injective resolution. To construct such resolution, we need to use a double complex, which may occupy the whole second quadrant $x<0, y>0$, assuming cohomology is used. How do we view it as a single complex?
I guess for the i-th piece, we take direct sum/product of all entries on the antidiagonal $x+y=i$. However it involves infinitely many terms, so direct sum and direct product are different. So which one is correct?