The most common and most likely answer to this crossword clue is the 13 letter word APPROPRIATELY
I've seen this crossword clue in Financial Times
A
P
P
R
O
P
R
I
A
T
E
L
Y
Buddy explains!
The straight part
We know that "in the correct way" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in the correct way = appropriately.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does APPROPRIATELY mean?
APPROPRIATELY adverb- In an appropriate manner; properly; suitably.