Assume we are given abelian categories $\mathcal{C}$, $\mathcal{D}$. We can form the (abelian?) category $\hom_{\mathrm{add}}(\mathcal{C},\mathcal{D})$ of additive functors from $\mathcal{C}$ to $\mathcal{D}$. What is the homotopy category $K^\mathrm{b}(\hom_{\mathrm{add}}(\mathcal{C},\mathcal{D}))$, in particular, what are its distinguished triangles?
Is it the case that any collection of natural transformations $F\Rightarrow G\Rightarrow H\Rightarrow F[1]$ such that $FX\to GX\to HX\to FX[1]$ is a distinguished triangle in $K^\mathrm{b}(\mathcal{D})$ for all $X∈K^\mathrm{b}(\mathcal{C})$ is a distinguished triangle in $K^\mathrm{b}(\hom_{\mathrm{add}}(\mathcal{C},\mathcal{D}))$? Dies the converse holds?