In the conclusion the first person is correct (5)
I believe the answer is:
emend
'is correct' is the definition.
(to emend something is to correct or improve it)
'in the conclusion the first person' is the wordplay.
'in the' indicates putting letters inside.
'conclusion' becomes 'end' (something's conclusion is its end).
'the first person' becomes 'me' (first-person pronoun).
'end' going around 'me' is 'EMEND'.
(Other definitions for emend that I've seen before include "do editorial work" , "Edit, improve" , "Make changes to (a text)" , "Correct/revise (a text)" , "Remove faults from" .)