Strolling Through Istanbul mentions that the nearby Byzantium city gate was known as the Pege, because of the nearby shrine of Zoodochus Pege, or the Life-Giving Spring. Several emperors built churches on this site, popular from early Byzantium times. The present one (part of the Balıklı Meryem Ana Rum Ortodoks Manastırı) is from 1833. The courtyard is covered with old tombstones inscribed in Karamanlı script, which is Turkish written in a Greek alphabet. Also in this courtyard are some tombs of bishops and patriarchs of the Greek Orthodox Church.
I took several pictures inside the church, as well as of the courtyard and its tombstones, that seem to refer to the profession of the ones who were buried at them.
After going down some steps one enters the small chapel with its ayazma or sacred spring. The Strolling guide mentions fish that lived there (with a story of how they have a brown side, having sprang from a frying pan into the water), but I did not see any.