I think that is actually the Benedictine Convent of Perpetual Adoration.
My cousin was the pastor there in the 1960s. By the early 1960s, when we were first living in Tucson, not enough young women were becoming nuns to make it a viable concern. I know that among the things they did was making the hosts for churches in Tucson.
The cousin was actually a cousin of my mother's, who'd become a priest in Ireland and was sent to America more or less as a missionary, and ended up in Tucson. He was the reason we wound up in that city when my mother decided to leave my alcoholic, abusive father, behind in northern New York State in 1962. She was a nurse, with five children still at home. She knew she could find work anywhere, and wanted to leave the snow and cold of NYS behind. She chose Tucson because of the family connection, as tenuous as it wa.
I have fond memories of attending Christmas midnight mass at the Convent in the mid 1960s. The convent is a gorgeous building, and it ought to be preserved in some way.
Calling it a monastery rather than a convent is to get its entire history wrong.