It is a prepositional phrase. 'Willing' is the noun object. The other words modify the noun.
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park sang joon I saw "them" as an object of "willing" and "to do so" as "objective complement.So I was wondering if you meant "(for) them to do so" is an adjective to-infinitive.Not quite:
park sang joonThank you, BillJ, for your very helpful answer.I know nothing about catenative complements, but I don't think "will" can take any complement.So I was still wondering the role of "to do so.""without [my willing them