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It has been shown recently that under certain conditions there exist dualities between different multiterminal (MT) source and channel coding problems. Following these results, we study lossless MT source coding and deterministic MT channel coding problems and point out different dualities between them. In particular, we show that there exists a functional duality between a Slepian-Wolf (SW) coding problem and a deterministic broadcast channel (DBC) coding problem and between a lossless multiple-description (MD) coding problem and a deterministic multiple-access channel (DMAC) coding problem. In analogy to the duality established between DBC and DMAC coding problems, we further propose a similar duality between SW and lossless MD coding problems; in this way, we form a closed "duality loop" of four MT coding problems, which imposes the existence of a single common rate point in the achievable rate regions of all four dual problems. We also consider duality in zero-error MT coding and shed light on practical code design with an example. Finally, extension to the case with only one lossless/deterministic component in the source/channel coding problem is provided.