This paper develops a common theoretical framework for understanding why climate emulators make errors. It connects pattern scaling, response functions, dynamic mode decomposition, and fluctuation-dissipation approaches through their treatment of the underlying dynamics. Simple box models and a modified Lorenz system isolate the effects of memory, hidden variables, noise, and nonlinear behavior. Response-function emulators perform best across the scenarios tested, while the analysis explains tradeoffs that broader applications must consider. Readers get practical implementation guidance and a basis for choosing an emulator and training experiments according to the sources of error they need to control.