The chant went up almost as soon as the game kicked off: “Alexis Sánchez, he wants to be blue.” This much seems to be true, though here was an awkward situation for the Chilean forward. Not quite in the shop window, as some would have it, because Manchester City are already known to be admirers. They would have signed him up by now, had Arsenal allowed it, and to say the player was frustrated by the non-events of the summer would be understating the case by quite a distance.

Sánchez has made his point on several occasions, yet this was not a time for the strop or the pained expression. Playing at the home of potential employers, Sánchez was content to let Mesut Özil do most of the grimacing, though he was clearly not enjoying watching City’s passing exhibition while barely an Arsenal pass of note was being aimed in his direction. What Sánchez had to do was look professional, and if possible lethal, just in case Pep Guardiola was beginning to think he no longer needs him.

It is even more debatable after this convincing win whether City do still need Sánchez in January, and just as difficult to see where he will fit into a forward line beginning to look menacing on a regular basis, because this victory, like the midweek one in Naples, was mostly achieved with Gabriel Jesus watching from the bench. Yet the same questions were applicable in the summer and Guardiola definitely wanted him then. He stands to be accused of over-egging the pudding, if not actual gluttony, yet it should probably be remembered that Jesus missed most of the second half of last season through injury, and it is also true that Wilfried Bony and Kelechi Iheanacho were moved on in the summer without being directly replaced. Sánchez can at the very least be viewed as an upgrade on either of those two, possibly both of them rolled into one.

Then there is the consideration that for all their confident first-time passing, their exemplary pressing and their lightning transitions from defence into attack, City are still not quite as clinical in front of goal as their manager would like them to be. Guardiola spent a lot of last season complaining that his side did not always score the goals their superiority in games deserved, and though to an extent the problem seems to have been addressed now that City top the scoring charts, they created enough chances here to be out of sight by the interval yet still ended up with a contest on their hands when Arsenal briefly threatened a second‑half fightback.

To use a favourite expression of Guardiola’s, City arrived many times in the first half but led only through Kevin De Bruyne’s dead-eyed opportunism. Certainly Sánchez would have fancied getting on the end of some of City’s sumptuous moves or exploiting some of the space the home side were creating. Sánchez would almost certainly have freed Leroy Sané to score, for instance, when Raheem Sterling’s woefully wayward pass put reports of his improvement this season into perspective. Sánchez might have had the nous to anticipate one of the David Silva crosses that rolled across the face of an unguarded goal, although in fairness Sergio Agüero was still on the pitch at the time, and he didn’t.

Sánchez stood up John Stones on one occasion, to set up a shot for Aaron Ramsey that Ederson saved, but it was a rare moment of effectiveness on an afternoon spent mostly in isolation. His patience was gradually worn down to the point where his temper got the better of him in picking up a 77th‑minute booking for a foul on Silva, possibly his way of registering annoyance at the fact that the City player had looked fractionally offside in supplying Jesus for the decisive third goal.

It cannot have been the easiest afternoon to endure, even for a player who has a habit of making most match‑day afternoons look only just about bearable, yet it could easily have been worse for the Chilean. Sánchez managed to give the ball away from a second-minute Arsenal corner, not normally a crime but these days an invitation for City to race the length of the pitch and set up a goal chance with just a couple of quick passes. Sterling and De Bruyne did just that, and Sánchez must have been relieved Agüero put his shot wide with a clear sight of goal, because going behind so early in such a fashion would have rivalled West Ham’s well-documented naivety in leaving themselves exposed at corners.

At the final whistle Sánchez appeared conflicted. His first, understandably diplomatic instinct was to get off the pitch as quickly as possible, then he realised his team‑mates were staying out to shake hands and swap shirts with their City counterparts. So Sánchez stopped and turned and stood apart, reluctant to join in but not wanting to appear aloof either. Caught in no man’s land, in other words, which is where he has spent most of the season.