Cokernel and Mapping Cone in Grothendieck Group

Question

I suppose this question is very easy, but at the moment I cannot find a rigorous answer.

Imagine there is an adjunction of exact functors $F:\mathcal C\stackrel{\dashv}{\longleftrightarrow} \mathcal D:G$, which has a natural transformation $\eta: 1\Rightarrow GF$ as adjunction unit. Now consider its cokernel $C=\operatorname{coker}(\eta): \mathcal C\rightarrow \mathcal C$. I want to know what is its action in the Grothendieck group $K_0(\mathcal C)$ but I cannot assume $C$ to be left-exact.

I can pass to the derived category $D^b(\mathcal C)$, where I can find a triangle $M\rightarrow GFM\rightarrow K\rightarrow$. Since I know an expression for $[GFM]\in K_0(\mathcal C)$, I can deduce what $[K]$ is.

My question: Is there any reason why $[K]$ should be actually $[CM]$?

Or: Are the the left derived functor of a cokernel of a nat. trafo and the mapping cones of the associated maps related in some way?

Thanks in advance!

Answer 1

What I found today which brings much more light into the topic (at least for me): The cokernel of morphisms in is a functor $\mathcal C^{[1]}\rightarrow \mathcal C$, whose derived functor is the mapping cone $D(\mathcal C^{[1]})\rightarrow D(\mathcal C)$. Since $\eta: \mathcal C\rightarrow \mathcal C^{[1]}$ induces $D(\mathcal C)\rightarrow D(\mathcal C^{[1]})$, which yields an endofuctor cone() of D().