I am having troubles completing the proof of theorem 6.8 (page 44) from Birger Iversen, Cohomology of Sheaves. (pdf here)
Previously we had constructed a functor $\rho$ from $K^+(A)$ (the homotopy category of bounded below complexes in $A$) to $D^+$ (The homotopy category of bounded below complexes of injectives in $A$).
The theorem says that $\rho$ transforms triangles into triangles. The last line of the proof says "It is easy to conclude the proof by means of 6.2", but i can't figure out how to do it. This is what I got so far:
$$\begin{array} $\rho X^\circ& \longrightarrow&\rho Y^\circ& \longrightarrow &Con^\circ(\rho f)& \longrightarrow& \rho X^\circ [1] \\ \downarrow{1}&&\downarrow{1}&&\downarrow{[\phi,1]^{-1}c}&&\downarrow{1}&&\\ \rho X^\circ& \longrightarrow&\rho Y^\circ& \longrightarrow &\rho Z^\circ& \longrightarrow& \rho X^\circ [1] \\ \end{array} $$
Where $\phi$ is the arrow that comes from "filling in the third arrow" (see proof of iversen). In order to prove that the lower line is a triangle we have to show that this diagram is homotopy commutative (which I managed to do) and that the vertical arrows are homotopy equivalences.
So my question is: "How do you prove that $[\phi,1]^{-1} c $ is a homotopy equivalence ?"
Or finishing the proof in another way would also help me, thanks in advance.