2 “In every place, both theirs and ours.” For although the letter be written to the Corinthians only, yet he makes mention of all the faithful that are in all the earth; showing that the Church throughout the world must be one, however separate in various places; and much more, that in Corinth. And though the place separate, the Lord binds them together, being common to all. Wherefore also uniting them he adds, “both theirs and ours.” And this is far more powerful [to unite], than the other [to separate].
For as men in one place, having many and contrary masters, become distracted, and their one place helps them not to be of one mind, their masters giving orders at variance with each other, and drawing each their own way, according to what Christ says, “You cannot serve God and Mammon;” so those in different places, if they have not different lords but one only, are not by the places injured in respect of unanimity, the One Lord binding them together. “I say not then, (so he speaks,) that with Corinthians only, you being Corinthians ought to be of one mind, but with all that are in the whole world, inasmuch as you have a common Master.”
This is also why he has a second time added “our;” for since he had said, “the Name of Jesus Christ our Lord,” lest he should appear to the inconsiderate to be making a distinction, he subjoins again, “both our Lord and theirs.”
Source: Homilies on First Corinthians (New Advent)