The Byzantine Agreement (BA) problem has been widely discussed in the literature concerned. Most BA protocols can reach a common agreement by way of fault masking. However, in a highly reliable fault-tolerant distributed system, just reaching a common agreement is not enough. We need to take into consideration another related problem called the Fault Diagnosis Agreement (FDA) problem. The goal of solving the FDA problem is to make each fault-free processor detect/locate the common set of faulty components in the distributed system. In this study, we shall solve the FDA problem with the dual failure mode over a multicasting network system. The proposed FDA protocol can detect/locate the maximum number of faulty components with the dual failure mode to solve the FDA problem in a synchronous multicasting network.