Polycarp testifies that Paul's teachings match the apostles.1
Ignatius testifies that Paul's teachings match the apostles.1
Clement of Rome testifies that Paul's teachings matched his own teaching. This is relevant Clement's teachings matched that of the apostles.
Contemporary Christians were saying that Paul “who once persecuted us is now preaching the faith which he once tried to destroy.” This is relevant because the faith that Paul was trying to destroy was apostolic Christianity (i.e. what the apostles were teaching).
Paul desired to have his teachings conform to the apostles' teachings.1 This is relevant because, if Paul wanted his teachings to conform to the apostles', it would have been very easy for him to do it.
But wait, couldn't it simply be that…
• …the Jerusalem churches' were not very well networked to Paul (i.e. their teachings were inaccessible to him), despite his desire?
We know this for everal reasons:
• Yale Anchor Bible Dictionary: “At the very least, one may say that Paul took these leaders seriously even if he did not understand them to be his ecclesiastical superiors.” [Vol. 5, 252.]
Richard Bauckham (NT professor): It is very notable that in Galatians, even in the context of Paul's strong concern to maintain the independence of his apostleship from Jerusalem, he admits that three years after his call to be an apostle he did visit Jerusalem and spent two weeks with Peter (Gal 1:18). [Jesus and the Eyewitnesses (Wm. B. Eerdmans, 2006), .]So:
F.F. Bruce: Paul was anxious both then and throughout his apostolic career to establish and maintain bonds of fellowship with the Jerusalem church and its leaders. [The Epistle of Paul to the Galatians: A Commentary on the Greek Text (Eerdmans, 1982) 99.]
Paul believed that his teachings matched the Apostles, and that they were on the same team. This is relevant because Paul would not believe this if his teachings did not actually match the apostles.
1 Cor 15:9, 11, 15 -- For I am the least of the apostles, … but I labored even more than all of them… Whether then it was I or they, so we preach and so you believed. … if Christ has not been raised, then our preaching is vain, …Moreover we are even found to be false witnesses of God, because we testified against God that He raised Christ
• Christopher Bryan (NT professor at Sewanee): According to Paul… that proclamation followed a basic pattern on which all were more or less agreed―a claim that could, as we have noticed, have been easily enough challenged if it were not true. [The Resurrection of the Messiah (Oxford, 2011), 66.]
• Christopher Bryan: The other is the appeal to what is new, to the autoptai, the eyewitnesses. This thing was not done in a corner. Some among the Corinthians were certainly familiar with the teaching of Cephas (1 Cor 1:12). Evidently they knew who James was and were aware of other apostles (15:8), and it is hardly likely that none among them had ever heard any of them teach. In other words, the assertion of eyewitness testimony made both by Paul and by the apostolic formula was easily open to challenge unless, as must have been the case, he and the Corinthians knew perfectly well that it was correct. (Cf. Robertson and Plummer, First Corinthians, 343; C.H. Dodd, "The Appearances of the Risen Chrsit: An Essay in Form-Criticism of the Gospels," in Studies in the Gospels: Essays in Memory of R.H. Lightfoot, ed. D. E. Nineham (Oxford, 1957), 30.; Wright, Resurrection of the Son of God, 318-19.) Naturally, the two elements do not serve the same purpose. The appeal to Scripture and antiquity is meant to tell us something of the significance of the apostolic testimony. The appeal to eyewitnesses is meant to tell us that the testimony is true. [The Resurrection of the Messiah (Oxford, 2011), 54.]
Rather than being disconnected, Paul cooredinated quite closely with the apostles and stayed in touch with them and their informants throughout his ministry.
• Paul spent a long amount of time talking to Peter about the historical Jesus and apostolic doctrine:
• James Dunn: “Paul and met and spoken with Peter and James on his first visit to Jerusalem (Gal 1:18f); he was on terms of close intimacy with several at least of 'the apostles' (Andoronicus and Junias - Rom 16:7; Barnabaus - Gal 2:9; 1 Cor 9:5f.; probably Silvanus - 1 Thes 2:6f.; and probably Apollos - 1 Cor 4:9); and he must have known not a few of the 'more than 500 brethren' (1 Cor 15:6), since it is probably Paul himself who has added the phrase, 'most of whom are still alive...'.” [98.]
• C. H. Dodd: “we may presume they did not spend all the time talking about the weather.” [The Apostolic Preaching and Its Developments 2nd ed.(Hodder and Stoughton, 1944). 16.]
The apostles at Jerusalem accepted Paul as a brother and co-worker.1
Paul was in the habit of submitting his teachings to the apostles This is relevant because the apostles would have corrected Paul's teachings if they were wrong.
• Acts 15: 1-23 -- Some men came down from Judea and began teaching the brethren, “Unless you are circumcised according to the custom of Moses, you cannot be saved.” 2 And when Paul and Barnabas had great dissension and debate with them, the brethren determined that Paul and Barnabas and some others of them should go up to Jerusalem to the apostles and elders concerning this issue. 3 Therefore, being sent on their way by the church, … 4 When they arrived at Jerusalem, they were received by the church and the apostles and the elders, and they reported all that God had done with them.… 6 The apostles and the elders came together to look into this matter. …12 All the people kept silent, and they were listening to Barnabas and Paul… James answered, …22 Then it seemed good to the apostles and the elders, with the whole church, to choose men from among them to send to Antioch with Paul and Barnabas—Judas called Barsabbas, and Silas, leading men among the brethren, 23 and they sent this letter by them, [saying] “…it seemed good to us, having become of one mind, to select men to send to you with our beloved Barnabas and Paul,…”
Paul and his churches sent financial relief to the Jerusalem church. This is relevant because his churches were the Gentile churches, and Paul, and it was he who who collected the funds for.
Peter disagreed with Peter over doctrine when Peter visited Antioch.1 This is relevant because it suggests Paul disagrees with Peter and James.
But so what? Plausibly, • They were still allies and in essential agreement.2 • Peter granted Paul's admonishment.