Bishop Ó Connmhaigh was one of at least two bearers of the surname (now rendered as Conway to hold this office.
An apparent kinsman, Máel Muire Ó Connaig, may have been a relative who bore an earlier version of the surname.
His immediate predecessor, Nicol Ó Duibhghiolla, had been appointed before October 1419 but never consecrated.
Ó Connmhaigh was appointed 23 October 1419; he had previously been abbot of the Cistercian Abbey of Corcomroe in the Burren.
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