As also ye have acknowledged us in part, that we are your rejoicing, even as ye also are ours in the day of the Lord Jesus.
In part. In contrast to "even to the end;" the testimony of his life was not yet completed (Theophylact). Rather, "in part" - i:e., some of you, not all (Grotius). So in 2 Corinthians 2:5; Romans 11:25. The majority at Corinth had willingly complied with Paul's directions in the first letter; but some were still refractory. Hence, arises the difference of tone in different parts of this letter. (See 'Introduction.')
That - `because' (Olshausen).
Your rejoicing - your subject of glorying.
Are - not merely shall be, implies the present recognition of one another as a subject of mutual glorying: about to be realized in its fullness "in the day (of the coming) of the Lord Jesus."