$\bar{d}$-continuity for countable state shifts

For full shifts on finite alphabets, Coelho and Quas showed that the map that sends a Hölder continuous potential $\phi$ to its equilibrium state ${\mu }_{\phi }$ is $\bar{d}$-continuous. We extend this result to the setting of full shifts on countable (infinite) alphabets. As part of the proof, we show that the map that sends a strongly positive recurrent potential to its normalization is continuous for potentials on mixing countable state Markov shifts.