  A detection system with a single sensor and two detectors is considered,
where each of the terminals observes a memoryless source sequence, the sensor
sends a message to both detectors and the first detector sends a message to the
second detector. Communication of these messages is assumed to be error-free
but rate-limited. The joint probability mass function (pmf) of the source
sequences observed at the three terminals depends on an $\mathsf{M}$-ary
hypothesis $(\mathsf{M} \geq 2)$, and the goal of the communication is that
each detector can guess the underlying hypothesis. Detector $k$, $k=1,2$, aims
to maximize the error exponent \textit{under hypothesis} $i_k$, $i_k \in
\{1,\ldots,\mathsf{M}\}$, while ensuring a small probability of error under all
other hypotheses. We study this problem in the case in which the detectors aim
to maximize their error exponents under the \textit{same} hypothesis (i.e.,
$i_1=i_2$) and in the case in which they aim to maximize their error exponents
under \textit{distinct} hypotheses (i.e., $i_1 \neq i_2$). For the setting in
which $i_1=i_2$, we present an achievable exponents region for the case of
positive communication rates, and show that it is optimal for a specific case
of testing against independence. We also characterize the optimal exponents
region in the case of zero communication rates. For the setting in which $i_1
\neq i_2$, we characterize the optimal exponents region in the case of zero
communication rates.
