A theory of how agents can come to understand a language is presented. If understanding a sentence $α$ is to associate an operator with $α$ that transforms the representational state of the agent as intended by the sender, then coming to know a language involves coming to know the operators that correspond to the meaning of any sentence. This involves a higher order operator that operates on the possible transformations that operate on the representational capacity of the agent. We formalize these constructs using concepts and diagrams analogous to category theory.