In the beginning, in a cave found in a valley near that mountain, there lived the itinerant holy father Neophytos, an ancient saint of Cyprus. Here he also had a small church in another cave, which has survived to the present, which is inside the monastery on the western side…
In the same place, where the cliff was collapsed, with difficult access like to a swallow’s nest, is found the cave of Saint Neophytos, of which I spoke earlier. In the same place is the church dedicated to the holy man*. It is very small and is covered with very strange icon paintings, unusual, terrifying, and calling every pious worshipper to repentance.
*the church in the cave is dedicated to the Holy Cross
Neophytos the Recluse, Typikon
“So from that time on [i.e., from 1170 onward] the structures of the Enkleistra began to be extended and adorned, and the entire length of the cliff was excavated for the construction of cells. And in the twenty-fourth year of my confinement  the Enkleistra was painted throughout and the cliff next to it was hewn out and consecrated as the church of the Holy Cross.”
Saint Neophytos the Recluse, Typikon, translation C. Mango – E. Hawkins, "The Hermitage of St. Neophytos and Its Wall Paintings", Dumbarton Oaks Papers 20 (1966) 124.
” … the Encleistra monastery, founded by St. Neophytos near Paphos, Cyprus in
1159, exhibits the pattern found in Russia and the West: its founder Neophytos went for pilgrimage to Jerusalem and the Holy Land and used his memories and imagination to recreate in his own hermitage near Paphos the secret places and loca sancta of Jerusalem.”
Natalia Teteriatnikov, “The Relic of the True Cross and Jerusalem Loca Sancta: The Case of the Making of Sacred Spaces in St. Neophytos’ Encleistra, Paphos”, in: New Jerusalems: Hierotropy and Iconography, ed. Al. Lidov, Moscow 2006, p. 409.
“In the innermost part of the cave he excavated a tomb for himself. He gave his hermitage the name of the Holy Cross and set up an altar in it so that he should not be deprived of holy communion. Five years later he set out quest of a particle of the True Cross and was fortunate in finding one … Neophytos must have obtained the particle in Cyprus, possibly from Stavrovouni.”
C. Mango – E. Hawkins, "The Hermitage of St. Neophytos and Its Wall Paintings", Dumbarton Oaks Papers 20 (1966) 123-124.