2 Cor 2:13-14
2 Cor 2:1-13 is written in the first person singular. In 2 Cor 2:12-13 Paul describes his anxiety at not finding Titus in the Troad.
2 Cor 2:14-3:3, on the other hand, is written in the first person plural. The 'we' here must refer to Paul and his co-sender, Timothy, since 2 Cor 3:1 concerns the writing of the letter. 2 Cor 3:1 refers back to 2 Cor 2:14-17 so the same 'we' is in view throughout.
So 2 Cor 2:12-13 concerns Paul and Titus, whereas 2 Cor 2:14-3:3 concerns Paul and Timothy. If Titus was not Timothy this is an abrupt switch and many commentators, working under the two-person assumption, fail to see a link between these sections.
However, if Titus was Timothy both sections concern Paul and Titus-Timothy. 2 Cor 2:12-13 concerns the Titus-Timothy's delay in Corinth and Paul's evangelistic success in the Troad and journey ot Macedonia. 2 Cor 2:14 continues the theme of the movements of Paul and Titus-Timothy. 2 Cor 2:15 refers to the work of Paul and Titus-Timothy respectively among 'those who are being saved' (which is a reference to the aforementioned converts in the Troad) and 'those who are perishing' (which refers to the opponents in Corinth, who probably caused Titus-Timothy's delay and would then be in view in 2 Cor 2:12-13). Thus Titus-Timothy allows a rather smooth reading of 2 Cor 2:12-3:1.
2 Cor 6:14-7:1
Michael Goulder showed that Paul makes the same points in the same order in 1 Cor 4-6, 2 Cor 6:4-7:1, and 2 Cor 10-13 ("2 Cor 6:14-7:1 as an Integral Part of 2 Corinthians" Nov Test 36, 1 1994 p47-57). The table below demonstrates this.
Some (sinners) criticize me,..1 Cor 4:3.........2 Cor 6:3........2 Cor 10:1-2
but I am a servant of God,.....1 Cor 4:1-5......2 Cor 6:4........2 Cor 10:7; 11:5,23
for I suffer tribulation...........1 Cor 4:6-13....2 Cor 6:4-10...2 Cor 11:23-33
I am your father and insist....1 Cor 4:14-21..2 Cor 6:11-13....2 Cor 12:14-18
you reform the sinners..........1 Cor 5-6........2 Cor 6:14-7:1..2 Cor 12:20-13:2
In each of these passages Paul responds to criticism by saying that he is a servant of God and has suffered much. He then says that he has fatherly affection for the Corinthians and tells them to put an end to the sin among them.
There is a logic to the structure of Paul's argument here. He must first address the criticisms and re-establish his authority before he can command the Corinthians to turn from sin. This explains the logical connection between 2 Cor 12:19 and 2 Cor 12:20 (notice the "for", γὰρ). A connection between the sin and the challenge to Paul's authority is also evinced by 2 Cor 13:2-3. Also note Paul's references to boasting and being puffed up in 1 Cor 4:6-7, 18-19 as well as in 1 Cor 5:2, 6.
Paul, therefore, makes essentially the same argument in all three passages. There are some important implications of this:
1) as Goulder points out, there is no need to see 2 Cor 6:14-7:1 as in interpolation, and the unfaithful in this passage may well refer to faithless Christians.
2) Nothing much has changed between the writing of 1 Corinthians and 2 Corinthians. This makes it unlikely that the interval between the two letters included a) a visit by Paul to Corinth, b) a letter of Paul to Corinth, c) the sending of Titus to Corinth, d) 18 months. Another day I will argue that the Titus's visit to Corinth (2 Cor 7) was one and the same as Timothy's (1 Cor 4:17; 16:10).
3) 2 Cor 10-13 repeats the line of argument found earlier in the letter, in much the same way as the subscription of Galatians (Gal 6:11-18) repeats the argument of Gal 1:1-10 and Gal 5:2-12 (see here). This confirms that 2 Cor 10-13 is indeed the subscription, written in Paul's own hand. This, in turn, supports the Titus-Timothy hypothesis and the unity of 2 Corinthians (see here).
2 Cor 9
I have argued here and here that the legality of the collection was open to challenge and that the collection was in danger of being intercepted. Paul would not have given information in his letter that would help opponents of the church intercept the collection or get it banned. This may explain why he writes in 2 Cor 9:1 that it was not necessary for him to write about the ministry to the saints. He may be referring only to sensitive information, such as the recipients of the collection, who are never named in 2 Corinthians. It is not necessary to suppose, as some do, that 2 Cor 9:1 could not have belonged to the same letter as 2 Cor 8.
The Titus-Timothy hypothesis (directly or indirectly) explains all the apparent discontinuities in 2 Corinthians and renders partition theories obsolete.