Build library outside, vertically (7)

I believe the answer is:

erectly

I'm a little stuck... Click here to teach me more about this clue!

'vertically' is the definition.
Both the answer and definition are adverbs. Maybe you can see a link between them that I can't see?

'build library outside' is the wordplay.
'build' becomes 'erect' (erecting is a kind of building).
'outside' suggests removing the centre.
'library' with its centre removed is 'ly'.
'erect'+'ly'='ERECTLY'

Can you help me to learn more?

(Other definitions for erectly that I've seen before include "In upright manner" , "In a rigidly upright way" .)

I've seen this clue in the Sydney Morning Herald.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.