Almighty God, faithful to His promise to Abraham and his children, sent His Son to save His people; while in His mercy He willed to redeem the heathen as well. Therefore Christ is the King whom, as its Redeemer, the whole world must hail and adore (Introit and Gradual).
In this miracle at Cana, a type of the Holy Eucharist, did our Lord formally manifest His divinity, i.e. His character as divine and therefore royal, and "His disciples believed in Him". The turning of water into wine is a type of transubstantiation, called by St. Thomas the greatest of all miracles, by which the wine of the Eucharist becomes the blood of the covenant of peace which God has made with His Church. Since also the divine King wishes to espouse our souls and since, as Bossuet says, it is through the Eucharist that this mystical marriage is consummated, the marriage feast at Cana also signifies the union of the Word with His spouse the Church.
"Having been invited to the wedding feast at Cana in Galilee," says St. Augustine, "our Lord attended, that being alone the author of the sacrament of Matrimony, He might confirm conjugal chastity." He also meant to make known to us the mystery of which these nuptials were the sign, that is, the union of Christ with His Church. For even those who, by a vow, have bound themselves to almighty God in the virginal state are not without nuptials since, with the whole Church, they have a part in the nuptials in which Christ Himself is the Spouse, and in this case, our Lord is typified by the bridegroom who kept to the end the good wine, that is, the Gospel.
Liturgical note: the Time after Epiphany begins on January 14 and lasts, so far as the Temporal cycle is concerned, until Septuagesima; and in the Sanctoral cycle until the Purification, February 2.
Source: Dom Gaspar Lefebvre, OSB, 1945, adapted and abridged.