Yes Elvis you’re the one who speculated John was alive when Ignatius wrote to the Smymaeans and the Catholic Encyclopedia speculates about a 10 yr gap inbetween. But of course Christ’s universal church existed. Catholic being defined as such.

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St. Ignatius may have been a disciple of St. John. The
Catholic Encyclopedia states:
More than one of the earliest ecclesiastical writers have given credence, though apparently without good reason, to the legend that Ignatius was the child whom the Savior took up in His arms, as described in Mark, ix, 35. It is also believed, and with great probability, that, with his friend Polycarp, he was among the auditors of the Apostle St. John.
The term is derived from Greek, as used in the New Testament.
**Acts 9:31
(IGNT+) αιG3588 THE μενG3303 INDEED ουνG3767 THEN εκκλησιαιG1577 ASSEMBLIES καθG2596 THROUGHOUT οληςG3650 WHOLE τηςG3588 THE ιουδαιαςG2449 OF JUDEA καιG2532 AND γαλιλαιαςG1056 GALILEE καιG2532 AND σαμαρειαςG4540 SAMARIA ειχονG2192 [G5707] HAD ειρηνηνG1515 PEACE, οικοδομουμεναιG3618 [G5746] BEING BUILT UP καιG2532 AND πορευομεναιG4198 [G5740] GOING ON τωG3588 IN THE φοβωG5401 FEAR τουG3588 OF THE κυριουG2962 LORD, καιG2532 AND τηG3588 IN THE παρακλησειG3874 COMFORT τουG3588 OF THE αγιουG40 HOLY πνευματοςG4151 SPIRIT επληθυνοντοG4129 [G5712] WERE INCREASED.
G2596
κατά
kata
kat-ah’
A primary particle; (preposition) down (in place or time), in varied relations (according to the case [genitive, dative or accusative] with which it is joined): - about, according as (to), after, against, (when they were) X alone, among, and, X apart, (even, like) as (concerning, pertaining to, touching), X aside, at, before, beyond, by, to the charge of, [charita-] bly, concerning, + covered, [dai-] ly, down, every, (+ far more) exceeding, X more excellent, for, from . . . to, godly, in (-asmuch, divers, every, -to, respect of), . . . by, after the manner of, + by any means, beyond (out of) measure, X mightily, more, X natural, of (up-) on (X part), out (of every), over against, (+ your) X own, + particularly, so, through (-oughout, -oughout every), thus, (un-) to (-gether, -ward), X uttermost, where (-by), with. In composition it retains many of these applications, and frequently denotes opposition, distribution or intensity.
G3650
ὅλος
holos
hol’-os
A primary word; “whole” or “all”, that is, complete (in extent, amount, time or degree), especially (neuter) as noun or adverb: - all, altogether, every whit, + throughout, whole.
kata holos
kat-ah’ hol’-os
Catholic is an adjective derived from the Greek adjective καθολικός (katholikos), meaning “universal”.**
There was only one Church during the time of the Apostles. Just because the term ‘Catholic’ means universal, does not take away from the fact that the universal Church was the Catholic Church. There were not many denominations then.