[T]he catholic Church of God is the koinonia of local churches mutually recognizing themselves as churches of God. This mutual recognition we think is essential. The Latin West concealed this in its desire to make everythng depend upon the relationship with the Church of Rome and its bishop. The catholic communion was seen as a totality of local churches all in communion with the sedes of Rome, without it being made clear that this necessary relationship with Rome is in the service of the mutual koinonia of local churches throughout time and space. In the gospel of God, which expresses the divine plan to reconcile all the human blocs shredded by sin, this mutual relationship is what counts more than anything else. What good would it be for them all to be in communion with Rome if the local churches remained water-tight compartments, shut up in their differences, as portrayed in a book for children which shows the Church as a great sun radiating around Rome, with the rays only converging. In the Holy Spirit and by the power of the Eucharist, it is mutual recognition that forms the concrete fabric of koinonia.[…]
The function of the local church of Rome and of its bishop must be understood in this perspective. It seems to us above all to be a ministry of recognition. Its principal task is that of ensuring the mutual recognition of the churches and basically the maintenance in each of them of the traits of the Church of Pentecost. Thus it is the guardian of communion, a communion which is realized in and by the local churches themselves, not imposed by some authority that transcends them. For communion is not realized around Rome, but thanks to Rome.
J. M. R. Tillard, “The Local Church Within Catholicity.” The Jurist 52 (1992): 448–54.