Assume to have an abelian category $\mathcal{A}$, and consider its derived category $\mathcal{D^b(A)}$). Let $F:\mathcal{D^b(A)}\rightarrow\mathcal{D^b(A)} $ be a functor between triangulated categories. Assume moreover given two objects $A,B$ of $\mathcal{D^b(A)}$ and fixed morphisms $\alpha : F(A)\rightarrow A$, and $\beta: F(B)\rightarrow B$.
Now we can define a map of abelian groups $Hom(A,B)\rightarrow Hom(F(A),B) $ by $\psi \mapsto \beta\circ F(\psi)-\psi\circ\alpha$.
I would like to know what can one say about this map; in particular I wonder if are there nice sufficient conditions for a morphism to be in the image of this map, maybe after assuming something more about the data.
Is this problem, or some specific instance of it, treated in the literature somewhere?
Thank you very much!
Edit:
As Fernando Muro pointed out this is too general. Let's more explicitly assume that the category has a tensor product and consider $F=C\otimes(-)$, with $C$ another object of $D^b(A)$. This would mean that we have a product from $C\otimes A$ to $A$ and from $C\otimes B$ to $B$, given by our fixed morphisms $\alpha$ and $\beta$. In this situation, the kernel of the map between the $Hom$ groups defined above would correspond to the compatible morphisms from $A$ to $B$.
My question regarding the image is motivated by the fact that I already have a chosen morphism from $A$ to $B$ with particular properties and I want to modify it to be compatible with the product structures; it turned out that I could find an element in $Hom(F(A),B)$ such that this is possible iff this is in the image of the morphism above.
My wish is that given an element $\phi$ of $Hom(A,B)$ one may define some 'obstructions' (as groups or cohomology classes or something), depending only on the product structures and this $\phi$, which decide if $\phi$ is in the image of the morphsim above or not.
I wrote the question in a more general form since it seemed to me interesting and because I am looking also for possible literature about this. Moreover, the answer should be completely abstract.
Remark: I am working in the derived category of sheaves of $\mathbb{Q}$-vector spaces over complex manifolds.
