We explore the duality between the Gelfand-Pinsker problem of channel coding with side information at the transmitter and the Wyner-Ziv problem of lossy compression with side information at the decompressor in the operational sense: whether a capacity-achieving encoder-decoder sequence achieves the rate distortion function of the dual problem when the channel decoder (resp. encoder) is the source compressor (resp. decompressor). We show that there exist capacity-achieving channel coding schemes that also achieve the rate-distortion function for the dual problem. However, this duality does not hold for every capacity-achieving channel coding scheme. In particular, we show that the original capacity-achieving encoder-decoder scheme of Gelfand-Pinsker operates far from the Wyner-Ziv rate-distortion function.