Many people find Irenaeus’s justification for a fourfold Gospel canon to be a real stretch, to put it mildly. Indeed, Irenaeus scoffs at the Valentinians when they resort to numerology such as seeing the 30 aeons as represented by Christ’s age of 30 (1.3.1; Irenaeus counters in 2.22.6 that Christ was actually 49 years old with John 8:47 “you are not yet 50 years old” as a prooftext!). However, if we enter sympathetically into Irenaeus’s argument, many of his analogies (world zones, winds, pillars) undergird the universality of the Church. So the acceptance of four Gospels is defended as the practice of the “catholic” Church, as opposed to groups Irenaeus perceives as schismatic “heretics” who privilege one Gospel (3.11.7 Ebionites with Matthew, Gnostic separationists with Mark, Marcionites with Luke, Valentinians with John). Irenaeus calls attention to the beginnings of each of the four Gospels as summing of the character of that particular narrative, in accordance with the Scriptures and symbolized by one of the four living creatures (cf. Ezekiel, Revelation).
It is not possible that the Gospels can be either more or fewer in number than they are. For, since there are four zones of the world in which we live, and four principal winds, while the Church is scattered throughout all the world, and thepillar and groundof the Church is the Gospel and the spirit of life; it is fitting that she should have four pillars, breathing out immortality on every side, and vivifying men afresh. From which fact, it is evident that the Word, the Artificer of all, He that sits upon the cherubim, and contains all things, He who was manifested to men, has given us the Gospel under four aspects, but bound together by one Spirit. As also David says, when entreating His manifestation,You that sits between the cherubim, shine forth.For the cherubim, too, were four-faced, and their faces were images of the dispensation of the Son of God. For, [as the Scripture] says,The first living creature was like a lion,symbolizing His effectual working, His leadership, and royal power; the second [living creature] was like a calf, signifying [His] sacrificial and sacerdotal order; butthe third had, as it were, the face as of a man,— an evident description of His advent as a human being;the fourth was like a flying eagle,pointing out the gift of the Spirit hovering with His wings over the Church. And therefore the Gospels are in accord with these things, among which ChristJesus is seated. For that according to John relates His original, effectual, and glorious generation from the Father, thus declaring,In the beginning was the Word, and the Word was with God, and the Word was God.Also,all things were made by Him, and without Him was nothing made.For this reason, too, is that Gospel full of all confidence, for such is His person. But that according to Luke, taking up [His] priestly character, commenced with Zacharias the priest offering sacrifice to God. For now was made ready the fatted calf, about to be immolated for the finding again of the younger son. Matthew, again, relates His generation as a man, saying,The book of the generation of Jesus Christ, the son of David, the son of Abraham;and also,The birth of Jesus Christ was on this wise.This, then, is the Gospel of His humanity; for which reason it is, too, that [the character of] a humble and meek man is kept up through the whole Gospel. Mark, on the other hand, commences with [a reference to] the propheticalspirit coming down from on high to men, saying,The beginning of the Gospel of Jesus Christ, as it is written in Esaias the prophet,— pointing to the winged aspect of the Gospel; and on this account he made a compendious and cursory narrative, for such is the propheticalcharacter. And the Word of God Himself used to converse with the ante-Mosaic patriarchs, in accordance with His divinity and glory; but for those under the law he instituted a sacerdotal and liturgical service. Afterwards, being made man for us, He sent the gift of the celestial Spirit over all the earth, protecting us with His wings. Such, then, as was the course followed by the Son of God, so was also the form of the living creatures; and such as was the form of the living creatures, so was also the character of the Gospel. For the living creatures are quadriform, and the Gospel is quadriform, as is also the course followed by the Lord. For this reason were four principal (καθολικαί) covenants given to the human race: one, prior to the deluge, under Adam; the second, that after the deluge, under Noah; the third, the giving of the law, under Moses; the fourth, that which renovates man, and sums up all things in itself by means of the Gospel, raising and bearing men upon its wings into the heavenlykingdom.(Adversus Haeresis 3.11.7, translated by Alexander Roberts, William Rambaut; for Ireneaus text preserved in Latin and later Greek excerpts see Ben C. Smith)
One of Ireneaus’s analogies to the fourfold Gospel is the four cherubim. Irenaeus seems to be harmonizing descriptions from Revelation 4:6-9 and Ezekiel 1:5-26 (on the question of whether Irenaeus is using a source here see T.C. Skeat, “Irenaeus and the four-gospel canon,” NTS 34  and response from Annette Yoshiko Reed, “EUAGGELION: Orality, Textuality and Christian Truth in Irenaeus’ Adversus Haeresis“ VC 56 ). So the Lion represents John’s high and confident Christology of the powerful Word, the human Matthew’s opening geneology of Jesus as son of David and Abraham, the calf Luke’s priestly character from its introduction of the priest Zechariah as the father of John the Baptist, and the Eagle Mark’s prophetic character with its first reference to Isaiah the prophet followed by the descent of the spirit in the baptism. Together, each image is essential to a fully orbed understanding of Christ’s identity. Other patristic writers were fond of this analogy, though they depart from Irenaus about which symbolizes which Gospel. I found this useful chart below in the classic commentary by Henry Barclay Swete, The Gospel According to St. Mark: The Greek Text with Introductory Notes and Indices (2nd Edition; London: Macmillan and Co, 1908), xxxviii. The chart features Irenaues, Victorinus of Pettau notes on the Apocalypse, Augustine (De cons. ev. 1.9), and the Pseudo-Athanasian Synopsis…