Bukhari

Hadith Commentary:
It is mentioned in well-known narrations that the Messenger of Allah (sallallahu alayhi wa sallam) set out for the Battle of Hunayn in the month of Shawwal, after the conquest of Makkah.
In this narration, it is stated that the Messenger of Allah (sallallahu alayhi wa sallam) undertook the journey for the Battle of Hunayn during Ramadan itself.
Therefore, the reconciliation is that the blessed journey began in Ramadan,
and its completion occurred in Shawwal.
The occurrence of the Battle of Hunayn in Shawwal is correct.
(Qastallani)

Hadith Commentary:
➊
From these narrations, it is understood that the Messenger of Allah (sallallahu alayhi wa sallam) set out for the Battle of Hunayn in the month of Ramadan, whereas the Battle of Hunayn did not take place in Ramadan but rather in the month of Shawwal.
Various answers have been given to this issue, but the answer of Muhibb al-Tabari appears to be the most plausible: that what is meant by the Messenger of Allah (sallallahu alayhi wa sallam) setting out towards Hunayn in Ramadan is that, upon hearing of the unrest caused by Hawazin, he made plans to set out towards them in Ramadan—that is, “setting out” refers to the intention to depart.
This usage is common in the Arabic language.
➋
Hunayn is a valley ten miles from Makkah al-Mukarramah.
The reason for this battle was that when the Messenger of Allah (sallallahu alayhi wa sallam) intended to leave Makkah al-Mukarramah to assist the tribe of Khuza‘ah, the tribe of Hawazin was informed that he was about to attack them.
They came to confront him at the market of Dhu’l-Majaz.
The Messenger of Allah (sallallahu alayhi wa sallam) continued on until he reached the valley of Hunayn on the night of Sunday, and then on Sunday, the middle of Shawwal, reconciliation was made with them.
(‘Umdat al-Qari: 12/264.)