He describes the ‘lower’ city (Acra) or the City of David, as having the ‘shape of a moon when she is horned’—ie, a crescent moon.
This describes the City of David area seen in the attached picture which I took recently from the south, whilst visiting Jerusalem. The crescent shape can be clearly seen and I have outlined it in the second picture with a yellow line for clarity. (This is where Abraham, arriving from Beer Sheva in the south, would have caught his first view of HaMakom--The Place to which God was directing him with Isaac...see Genesis 22:4 'Then on the third day Abraham lifted his eyes and saw The Place [HaMakom] afar off').
The Valley of the Cheesemakers is not as discernible today for two reasons:
1. As Josephus informs us, the Hasmoneans ‘filled up that valley with earth.’
2. It is today covered with dwellings and other buildings
But the reason the Hasmoneans filled it is of great import: Josephus tells us that the reason was that they ‘had a mind to join the city to the temple.’
Clearly this can only mean that the temple was in this Acra region and the Hasmoneans desired to make the temple more easily accessible from the western side of the valley.
Ergo, the temple was in the Acra, or the portion recognised to ancients and today, as the City of David.