Allegri was hired to replace fellow Italian journeyman, Antonio Conte, at the end of last season, with the mission to continue his predecessor’s good work.
It’s not been the happiest start for Allegri, however, with Napoli sitting down in 10th with one win and two defeats from their three Serie A games so far.
Napoli were 2-0 up against reigning champions Inter at the Giuseppe Meazza eight minutes after the break and seemed to be on their way to an impressive three points.
Inter forwards Marcus Thuram and Lautaro Martinez had other ideas, however. Martinez pulled one back three minutes after Napoli’s second and Thuram equalised in the 82nd.
Martinez scored the winner in added time, capitalising on some chaos in the Napoli box to slot home from close range.
Speaking to reporters after the game, Allegri was clearly unhappy with his side, saying: "For the spectators it was a fun match, for us a bit less.
"We were up 2-0, then we conceded a goal that was too light and Inter changed their rhythm, creating difficulties for us.
“On the decisive goal we thought the ball had gone out and we stopped, but football is like this: we could have scored, they did."
"It remains a good performance, especially in terms of ball management," Allegri explained when dissecting the tactical aspects of the loss.
"In this moment we are paying too much for every mistake: we have conceded five goals in two games and we must improve, without thinking that the solution is only to defend inside the area.
“We are only at the third matchday, the condition will grow. Now we must stay calm and think about the Champions League."