Making small cocktail of martini, using bottled Asti at the end (13)

I believe the answer is:

miniaturising

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'making small' is the definition.

'cocktail of martini using bottled asti at the end' is the wordplay.
I cannot quite understand how this works, but
'at' is found in the answer.
This may be the basis of clue (or it may be nonsense).

Can you help me to learn more?

(Another definition for miniaturising that I've seen is " Making smaller".)

I've seen this clue in The Telegraph.

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