I am learning about derived categories and I'm confused with the definition of the derived functor that I've been given. Before stating it, I will set up some notations. Let $F:\mathcal{A}\to\mathcal{B}$ be an additive functor between abelian categories. Denote $K(\mathcal{A})$ to the homotopy category of chain complexes with terms in $\mathcal{A}$. Denote $K(F):K(\mathcal{A})\to K(\mathcal{B})$ to the induced functor between homotopy categories and call $K_\text{qis}(\mathcal{A})$ to the wide subcategory of $K(\mathcal{A})$ whose morphisms are the quasi-isomorphisms. Suppose $D(\mathcal{A})$ is the localization of $K(\mathcal{A})$ with respect to the quasi-isomorphisms and denote $Q_A:K(\mathcal{A})\to D(\mathcal{A})$, $Q_B:K(\mathcal{B})\to D(\mathcal{B})$ to the localization functors.
The following is the definition I've been given: we say that $RF:D(\mathcal{A})\to D(\mathcal{B})$, the total right-derived functor, is defined at a complex $A^\bullet\in K(\mathcal{A})$ if the diagram $$ \begin{align} A^\bullet/K_\text{qis}(\mathcal{A})=\{A^\bullet\xrightarrow{\text{qis}}I^\bullet\}&\to D(\mathcal{B})\\ (A^\bullet\to I^\bullet)&\mapsto (Q_B\circ K(F))(I^\bullet) \end{align} $$ is almost constant, where a diagram $G:I\to\mathcal{C}$ is said to be almost constant if there exists a cofinal and full subcategory $J\subset I$ (meaning the inclusion functor $J\to I$ is cofinal) such that for all morphisms $f\in J$ we have that $Gf$ is an isomorphism.
On that case we define $$ RF(A^\bullet)=\underset{\{A^\bullet\xrightarrow[\text{qis}]{}I^\bullet\}}{\text{colim}}(Q_B\circ K(F))(I^\bullet). $$
I am confused in several ways with this definition. The first thing is that I don't understand why "being almost constant" seems to be a sufficient condition for the existence of the colimit. We can indeed change the index category to the cofinal subcategory for computing the colimit, but I don't think it is true in general that a diagram in which all morphisms are isos must have a colimit. The intuitive approach to show this is to pick any object in the diagram, but what is not clear is the choice of "legs" that will turn that object into a cocone of the diagram.
Also, this only defines the action on objects of $RF$. What should the action on morphisms of $RF$ should be?
I've tried to look up the definition of total right derived functor on books and on the internet, but all of them seem different to the one I've given here. Does anyone know if this definition is equivalent to any of those available in the literature?
On the other hand, when trying to look on the web for references on "almost constant functors", the most similar concept I've been able to find is that of "essentially constant diagram" (defined, for example, here, which seems to guarantee existence of colimit). But these two notions seem different to me.