Just as Spock would keep anyone (and everyone) informed of his progress where The City was concerned. If there was any progress to inform them of, that is. It was difficult not to become too disconcerted. Were he fully Vulcan, he would not be swayed by defeat. But Spock was half Human, and that made his fight with himself all the more difficult.
He was not accustomed to failure. On the contrary, Spock had excelled beyond his means (and the means of others) in order to prove that he was capable of the best that Vulcan had to offer. Even without the other half of their genes.
"If you are safe to continue on your own, then I shall make my departure. My ship has been without my care for some time and I wish to ensure that it is still operating at optimal conditions."