Happy in company, the Italian supplies part of roof (6)

I believe the answer is:

cupola

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'part of roof' is the definition.
'cupola' can be an answer for 'roof' (cupola is a kind of roof). I'm not certain of the 'part of' bit.

'happy in company the italian' is the wordplay.
'happy' becomes 'up' (I've seen this in other clues).
'in' means one lot of letters goes inside another.
'company' becomes 'co' (abbreviation).
'the italian' becomes 'la' ('the' in Italian).
'up' put into 'co' is 'cupo'.
'cupo'+'la'='CUPOLA'

'supplies' is the link.

Can you help me to learn more?

(Other definitions for cupola that I've seen before include "Round domed roof" , "fireplace" , "Rounded dome" , "Dome - furnace" , "St Peter's roof" .)

I've seen this clue in The Irish Times.

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