Let $A$ and $B$ be abelian categories and let $F : A \to B$ be an additive functor. Let $K^+(F) : K^+(A) \to K^+(B)$ be the induced functor on the corresponding homotopy categories of left bounded complexes. Let $\pi_A$ and $\pi_B$ be the projections from the homotopy categories to the derived categories. Now, ideally, one would like to find a functor $T$ which makes the following diagram commute:
$$ \require{AMScd} \begin{CD} K^+(A) @>K^+(F)>> K^+(B)\\ @V\pi_AVV @VV\pi_BV \\ D^+(A) @>>T> D^+(B) \\ \end{CD}$$
But this is too much to expect, because $K^+(F)$ doesn't always carry quasi-isomorphisms to quasi-isomorphisms. So, one looks for the best thing that one can do to remedy this. There are two natural choices:
Among all pairs $(T,s)$ of functors $T$ together with a natural transformation $s: \pi_B \circ K^+(F) \to T \circ \pi_A$ one looks for an initial object, and if it exists, calls it the right derived functor. OR,
Among all pairs $(T,s)$ of functors $T$ together with a natural transformation $s: T \circ \pi_A \to \pi_B \circ K^+(F)$ one can look for a final object, and if it exists, call it the right derived functor.
Typically, one goes with the first choice for the definition of the right derived funcor.
Here's my question: Why is the second choice bad? I suspect that there is some degenerate candidate for the second choice, but I cannot find it.
Edit: I am aware that for left derived functors, one takes the second choice. But in that case one works with the categories of bounded above complexes. So my question is: why is choice 1 the correct choice for left bounded complexes and choice 2 the right one for bounded above complexes?
