The problem of sequential change diagnosis is considered, where a sequence of independent random elements is accessed sequentially, there is an abrupt change in its distribution at some unknown time, and there are two main operational goals: to quickly detect the change and, upon stopping, to accurately identify the post-change distribution among a finite set of alternatives. The algorithm that raises an alarm as soon as the CuSum statistic that corresponds to one of the post-change alternatives exceeds a certain threshold is studied. When the data are generated over independent channels and the change can occur in only one of them, its worst-case with respect to the change point conditional probability of misidentification, given that there was not a false alarm, is shown to decay exponentially fast in the threshold. As a corollary, in this setup, this algorithm is shown to asymptotically minimize Lorden's detection delay criterion, simultaneously for every possible post-change distribution, within the class of schemes that satisfy prescribed bounds on the false alarm rate and the worst-case conditional probability of misidentification, as the former goes to zero sufficiently faster than the latter. Finally, these theoretical results are also illustrated in simulation studies.