Answer
Paul asserted the status of his own writings as Scripture.
Explanation
God is the real author of the Scriptures and as such He decides when and where to begin and end His writings.
He never asked any of the Patriarchs to write any of His words. The first command to write is given to Moses by God:
“And Jehovah said to Moses, Write this, a memorial in a book”
(Exo 17:14).
So Moses began the Scriptures of God.
God prophesied in advance when and where He will end His Word:
“Bind up the Testimony (New Testament), seal the Law (Old
Testament) among My disciples (Isaiah 8:16).
Scripture Ended by the Apostles
The Scripture was to be bound up and sealed (completed) by the disciples of Yahweh. Who was this Yahweh? Let us see.
“Sanctify Jehovah of Hosts Himself, and let Him be your fear; and
let Him be your dread. And He shall be for a sanctuary, and for a
stone of stumbling, and for a rock of falling to the two houses of Israel; for a trap and for a snare to the ones living in
Jerusalem. And many among them shall stumble and fall, and be
broken, and be snared, and be taken. Bind up the Testimony, seal the
Law among My disciples” (Isaiah 8:13-16).
Both Peter and Paul quoting Isaiah’s prophecy, identify who this Yahweh of Hosts is:
“But to disobeying ones, He (Jesus) is the "Stone which those
building rejected; this One became the Head of the Corner, and a
Stone-of-stumbling, and a Rock-of-offense" to the ones stumbling, being disobedient to the Word, to which they were also
appointed” (1 Pet 2:7-8).
“For they stumbled at the Stone-of-stumbling, as it has been
written, "Behold, I place in" "Zion a Stone-of-stumbling," "and a
Rock-of-offense," "and everyone believing on Him will not be shamed” (Rom 9:32-33).
Yes, both Peter and Paul identified the Yahweh of Hosts as Jesus Christ. This is undeniable!
So, it was Jesus’ disciples who were predestinated to complete and bind up the Scriptures. Thus, bind up and seal the Scripture (both OT/Law and NT/Testimony of Jesus) was to be done by the disciples of Jesus.
Peter and Paul Knew This
Both Peter and Paul knew from prophecy through the Holy Spirit that they were to complete and bind up the Scripture.
Peter identifies Paul’s epistles as Scripture:
“as also our beloved brother Paul wrote to you, according to the
wisdom given to him; as also in all his epistles, speaking in them
concerning these things, in which are some things hard to understand,
which the unlearned and unsettled pervert, as also they do the rest
of the Scriptures, to their own destruction” (2 Pet 3:16).
“Rest of the Scripture” means clearly that Peter considers Paul’s writings as Scripture.
Paul identifies Luke’s gospel as Scripture:
“For the Scripture says, "You shall not muzzle an ox treading out
grain," and, “the laborer is worthy of his pay” (1 Tim 5:18).
Here Paul considers Deuteronomy 25:4 and Luke 10:7 (laborer’s pay) on an equal footing. This clearly shows that Paul considered Luke as Scripture!
Peter Arranges New Testament
“I think it only right for me to stir up your memory of these matters
as long as I am still alive. I know that I shall soon put off this
mortal body, as our Lord Jesus Christ plainly told me. I will do my
best, then, to provide a way for you to remember these matters
at all times after my death” (2 Pet 1:13-15; GNB).
Peter, knowing that he, as a disciple of the Lord of Hosts, was to “seal the OT” and “bind up the NT” says that he will make sure that the true Christians will have always “a way” to remember the true faith.
Paul Compiles the New Testament and Entrusts Timothy
“Only Luke is with me. Get Mark and bring him with you,
because he can help me in the work. When you come, bring my coat
that I left in Troas with Carpus; bring the books (OT) too, and
especially the ones made of parchment (the New Testament writings)” (2 Tim 4:11, 13).
There is an “urgency” in this command to Timothy. Luke is already with Paul. He asks Timothy to get Mark and bring him with Timothy to finish the work.
Second epistle of Timothy is Paul’s last epistle. He says he is about to be “poured out” (verse 6). So, as another disciple of the Lord of Hosts, he urgently needs to compile the NT. He already knows that Peter has edited the NT books. Now he wants to finish the final editing and entrust the same to Timothy before his death.
The Greek word for “coat” is “phelones”. Adam Clarke commentary:
“Τον φελονην is by several translated bag or portmanteau; and it
is most likely that it was something of this kind, in which he might
carry his clothes, books, and travelling necessaries.”
Yes, Paul was not worried about his cloths at his last breath. He was concerned about the bag with the OT books and the NT parchments he left with Carpus at Troas.
The “books” are definitely the OT books. What about the parchments or “membrana”. The parchments were made of sheep skin and were very expensive in those days. These were used only for very important documents.
In the first century the New Testament writings were made on parchments. These are the documents he meant by parchments. He made the final editing of the NT together with Luke and Mark after Peter had finished his part. Both the OT and NT were entrusted to Timothy for sacred use.
Conclusion
Paul knew about the Isaiah prophecy of the completion of OT and NT among the disciples of Jesus Christ.
He, as part of the disciples of the Lord, knew he was writing the Scripture. Hence he charges:
“I charge you by the Lord that this letter be read to all the
holy brothers” (1 Thess 5:27).
“And when this letter is read before you, cause that it be
read also in the Laodicean assembly, and that you also read the one
of Laodicea” (Col 4:16).
“But if anyone does not obey our Word through the letter, mark
that one, and do not associate with him, that he be shamed” (2 Thess
3:14).