Let me first say that my knowledge about Al-Azhar, beyond what I have already alluded to, is meagre. But when we talk about Salah al-Din in Egypt we need to distinguish between Salah al-Din the commander/governor of Egypt and Salah al-Din the ruler of Egypt. These are two different instances. In the first instance, Salah al-Din was following the commands of Noor al-Din Zangi, who was staunchly Sunni. However, where Noor al-Din Zangi showed less affinity for the Fatimids (or other Shias), Salah al-Din largely appears sympathetic or, at least, accomodating. To this extent, he defied Noor al-Din Zangi's orders to issue a declaration of abrogation of Fatimid Khilafat during Friday prayers after Egypt had completely fallen under the Zangi rule. This is where the gulf between Noor al-Din Zangi and otherwise very loyal Salah al-Din started to appear and widen. The deed was then assigned to an unknown person who made the said announcement on Noor al-Din Zangi's behalf after one fine Friday prayers. Salah al-Din had struck a friendship with the young but critically ailing last Khalifa of the Fatimids and ordered everyone not to inform the Khalifa of what had happened after the prayers and to let him die thinking he was the Khalifa till the end. If the conversion of Al-Azhar to Sunni fiqah took place during this period it might have been done so on the orders of Noor al-Din Zangi. As I said, keeping the abrogation of Fatimid Khilafat aside, Salah al-Din was loyal to Noor al-Din Zangi and took orders from him. But this is purely speculative and I have not come across any evidence for this.
In the second instance, the governing authority over Egypt by default fell to Salah al-Din over time. Even with the worsening of relations between the two of them over the abrogation issue Noor al-Din Zangi did not replace Salah al-Din. A short time later, Noor al-Din Zangi died and the rulership over his kingdom that included most of the major settlements of the Middle East, including the three Holy cities, fell to Salah al-Din. Outside of Wikipedia, I have not read any conversions (forced or voluntary) to Sunnism of the Egyptian population and Egyptian institutions by Salah al-Din, including the conversion of Al-Azhar to Sunni fiqah and burning of its books/written material (Shia and/or secular), both during his commandership/governorship and absolute rule. There are a lot of distorted narratives here thanks to the conflicting nature of our religious sects. I will need a little time to dig out information on how Al-Azhar became a predominantly Sunni institution. But in any case, its foundations - undoubtedly and undisputedly - have been laid down by the Fatimids, who were Ismaili Shias.