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Source coding under a rate-distortion (R-D) constraint and channel coding under a channel cost constraint have long been considered as duals of each other starting from Shannon (1959). We address this duality with the objective of providing a mathematical characterization of it in a functional sense. To be specific, given an optimal source (channel) coding scheme, we detail the conditions under which such a scheme is a functional dual to a channel (respectively source) coding scheme. We show that this functional duality is enabled through appropriate choices of metrics and constraints for the dual source (or channel) coding problem. We then extend this duality formulation to source and channel coding with side information.