A simple proof of the coding theorem for the multiple-access channel (MAC) with arbitrarily correlated sources (DMCS) of Cover-El Carnal-Salehi, which includes the results of Ahlswede for the MAC and of Slepian-Wolf for the DMCS and the MAC as special cases, is first given. A coding theorem is introduced and established for another type of source-channel matching problem, i.e., a system of source coding with side information via a MAC, which can be regarded as an extension of the Ahlswede-Körner-Wyner type noiseless coding system. This result is extended to a more general system with several principal sources and several side information sources subject to cross observation at the encoders in the sense of Han. The regions are shown to be optimal in special situations. Dueck's example shows that this is in general not the case for the result of Cover-El Gamal-Salehi and the present work. In another direction, the achievable rate region for the module-two sum source network found by Körner-Marton is improved. Finally, some ideas about a new approach to the source-channel matching problem in multi-user communication theory are presented. The basic concept is that of a correlated channel code. The approach leads to several new coding problems.