Since the unity that Christ desires His followers to have is exemplified in the unity of the three Persons of the divine Trinity (cf. John 17:22), it is important to think carefully about the nature of that divine unity.
The Christian doctrine of God is monotheistic. We believe in one God. "Hear O Israel: The LORD our God, the LORD is one." (Deut 6:4) The doctrine of the Trinity has to be understood upon that foundation. The three Persons of the Trinity are each God, and yet there are not three Gods, but one God. (cf. the Athanasian Creed) The Father and the Son and the Spirit are one in Being, three in Person.
So the Persons of the Trinity are not merely united in what they believe. Nor are they merely functionally united in working together on the same projects toward the same goal. Nor are they merely united in loving one another. They are united in all three of those ways, but their unity is greater than all three of those forms of unity. Three separate beings could be united in all three of those ways. The unity of the three Persons of the Trinity is fundamentally a unity in Being; they are one Being.
Because the unity Christ desires for His Church is to be like the unity He has with the Father, therefore the unity of the Church cannot merely be that of doctrine and function. The Church must also be one in being. Christ founded a single, visible Church, the one He mentions twice in St. Matthew’s Gospel. There were established, of course, in the spreading of the gospel local churches in the various cities, but always still one visible Church, with all her members in full communion with one another, through the unity of their respective bishops. If a person was in full communion in a local church, and if that person traveled to another city, he could receive the Eucharist in that local church. That is part of the reason why the Church was called Catholic; it was the one visible Church that was spread all over the known world. (We see this discussed in the fathers, and in St. Augustine. Travelers to another city had to be able to determine which assembly in that city belonged to the true Church, as opposed to some heretical sect, hence the very early development of the four ‘marks’ of the Church.)
What was the ground of the one Church's unity? It was not merely doctrinal agreement on essentials, although they had doctrinal agreement on essentials. Doctrinal agreement on essentials was what in philosophy we call an ‘essential accident’; it always and necessarily accompanies full communion, but is not itself the ground or basis of full communion. The ground of the full communion of these local churches was proximately their acceptance of the authority of the Apostolic churches from which they had been birthed, and specifically the leadership therein. And the ground of the full communion of the Apostolic churches was also rooted in their Apostolic origin, namely the full communion of the Apostles with one another, a full communion that was, after Christ’s ascension, grounded in the keys given to Peter. Thus the Church grew and spread in exactly the same way that an organic body grows, organically. And the unity of the Body depends on all its part remaining in organic unity with that from which it started, i.e. the Apostolic churches, and thus the Apostles, and thus Peter. That is why Apostolic succession is so important. Gnosticism cannot support full communion. The contemporary ecumenical movement's treatment of collaborative unity as the ideal treats the Church like an artifact wherein the parts make up a whole by virtue of performing a unified function. But artifacts do not have the same kind of unity had by organisms. Man can make artifacts, but only God can make organisms. Artifacts have only extrinsic unity; organisms have intrinsic unity (which is a much greater unity).
And the Church is a Body, an organism, and thus has an organic unity, not merely a functional unity. Organic unity is always historically traceable to the origin of the organism in the creative act that only God can do. That is why the Church cannot be a Body without Apostolic succession. Christ formed an organism that is the Church. We can have full communion with that organism only by being incorporated into that organism, not merely by collaborating and/or agreeing about some essential doctrines. Such artifactual unity pales in comparison to the organic unity Christ gave to His Body, the Church.
All the parts of an organism are by their very nature parts of that organism. That is not the case with artifacts. Think of Aristotle’s example of the wooden bed. If a wooden bed happens to get stuck in the soil, and the wood still has life in it, what sprouts up is not a bed, but a tree. Treating full communion with the Church as if it does not require incorporation into the Church treats the Church as an artifact, since collaborative unity does not require that the parts change their nature so as to become in their nature parts of that whole. But since the Church is not an artifact but an organic Body, full communion with the Church requires organic incorporation into the Church.
"Let unity, the greatest good of all goods, be your preoccupation." - St. Ignatius of Antioch (Letter to St. Polycarp)