Postnikov inverse and the specified nonconnective degree-zero group

The assignment $[M]\mapsto [M[0]]$ defines a map from $G_0({\mathcal{A}}_T)$ to the triangle group by the short-exact/cofiber assertion in [F1]. The Euler composite is the identity on each generator $[M]$ by [F1].

For $E\in {\mathcal{C}}_T$, choose $a\le b$ outside which its cohomology vanishes. Its actual truncation triangles ${\tau }_{\le n-1}E\to {\tau }_{\le n}E\to H^n(E)[-n]$ all lie in ${\mathcal{C}}_T$ by [F1]. Starting with ${\tau }_{\le a-1}E=0$ and ending with ${\tau }_{\le b}E\simeq E$, their relations give $[E]={\sum }_n(-1{)}^n[H^n(E)[0]]$; the sign follows from the triangle $A\to 0\to A[1]$. Hence the reverse composite is also the identity.

Apply [F2] to a small equivalent presentation of the idempotent-complete ${\mathcal{C}}_T$ and compose its inverse with the triangle Euler isomorphism just proved. This gives exactly the displayed ${\pi }_0$ isomorphism and object-class inverse for the current nonconnective functor. Take $T=U$ for the global comparison. Inclusion of the supported category is an actual exact functor preserving the same cohomology; naturality in [F2] and the Euler compatibility in [F1] give the asserted commutative support/global square. Positive-degree comparison alone is not used.