Remark 6.60.
The functor \(\tau_*\) admits a right adjoint \(\lim_n C_{\leq n} \to C\) given by sending a system \((X_n)\) of \(n\)-truncated objects to their limit \(X = \lim_n X_n\). We see:
\(\tau_*\) is fully faithful if and only if the unit map \(X \to \lim_n \tau_n X\) is an isomorphism for all \(X \in C\), i.e. every object is the limit of its Postnikov tower. Note that this is necessary but not sufficient for \(\tau_*\) to be an equivalence.
\(\tau_*\) is essentially surjective if and only if every system \((X_n)\) satisfying the compatibility conditions arises as \((\tau_n X)\) for some \(X \in C\).