No, because it is refers to the subject, νεανίας. The genitive article is for εἶναι, since ἐφίεμαι takes the genitive of the things aimed at (being wise, in the present case). In the nominative it would be τὸ σώφρων εἶναι. And you do need εἶναι, as you need being in the English.
It takes the genitive of τὸ σώφρων εἶναι, hence τοῦ σώφρων εἶναι. If it took the dative it would be
τῷ σώφρων εἶναι. σώφρων must remain nominative as it refers to the subject of the principal, νεανίας.
I still do not understand the answer. It seems to violate the rule that ἐφεῖτο takes a genitive, yet the answer adds an infinitive rather than using σώθρονος ἐφεῖτο. I don’t get it.
τοῦ εἶναι (genitive articular infinitive) is the object of ἐφεῖτο. σώφρων is the predicate nominative referring to the subject νεανίας. Maybe if we paraphrase:
το ειναι is a neuter nominative/accusative substantive, meaning “the act of being” or similar. του ειναι is a genitive substantive, τωι ειναι is a dative substantive.
As others have mentioned, it is an example of the articular infinitive (which is something I find quite hard when it comes up).
In the phrase -
‘τοῦ σώφρων εἶναι’ (of being wise)
it is worth looking at it from the point of view of predicate or attributive position (something I think Mastronarde isn’t clear on with articular infinitives).
So the expression is one unit, with the modifier (σώφρων) in the attributive. The unit as a whole is in the genitive, to go with ‘ἐφεῖτο’.
Another way to say it in English which would be fairly literal would be ‘That youth is desirous of being wise’.
I don’t think my explanation is very clear (and is also possibly wrong!), so hopefully someone else can express this better than I can.
I also am also not sure if articular infinitives come up enough to be worth your time researching Smyth or similar just yet, but maybe brushing up on the attributive/predicate positions might help (something which I always find to be a useful waste of my time).
At the risk of muddying the water still further, I think the confusion occurs in the process of translation.
If I were to translate the Greek literally as: ‘That the young man he aims of the to be chaste’, would that make it any clearer as to why the article/infinitive goes with the verb and the adjective goes with the noun?