You do, of course, realize that reading the modern self-designation of the Latin church back into St. Ignatius’ use of the phrase he katholike ekklesia in his Epistle to the Smyrnaeans is tendentious.
St. Ignatius’ own ecclesiology, as summed up in the very passage in which that phrase is used, “where the bishop is, there is the catholic church”, accords with the Orthodox view of the fundamental equality of bishops, which finds its fullest expression in the use in Slavonic of a single word, “sobornosty” to encompass both catholicity and conciliarity, rather with than the papal claims advanced by you Latins, which would seem to apply the modifier “of Rome” to the word bishop.
As such, his use of the phrase “Catholic Church” is arguably in agreement with the usage found in the Encyclical of the Eastern Patriarchs (1848), which expresses the consciousness of the Orthodox, that we constitute the One Holy Catholic and Apostolic Church.
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Of course, the Orthodox are part of the One Holy Apostolic Catholic Church, along with the Catholics, Orientals and Assyrians