The genitive signals a possessor as a predicate NP.
The genitive of the possessor as a predicate NP with εἰμί may indicate a person whose nature, duty, habit etc. it is to do that which is expressed in the infinitive which is the subject of the verb.
It is often preferable to use a more specific verb than 'to be' when translating: 'to belong to' 'to be suitable for' etc.
In this construction the possessor is generally human and definite. The possessor is the focus and thus represents new information for the listener. The subject is the topic and is already known.
A possessive pronoun may stand in the place of the genitive.
The construction εἰμί + genitive is distinguished from εἰμί + dative by the fact that the possessive relationship is always granted by nature or by a social convention (through legal ownership, for instance, or moral duty).
The genitive with a copula is the least frequent of the predicative constructions which indicate possession. However, as a consequence of its specialised semantic force it remains stable throughout the history of the language (Benvenuto 2014).
οὐ Κορινθίων τοῦ δημοσίου ἐστὶ ὁ θησαυρός, ἀλλὰ Κυψέλου τοῦ Ἠετίωνος
κλεπτῶν γὰρ ἡ νύξ, τῆς δ᾽ ἀληθείας τὸ φῶς.
τῆς γὰρ ἀρετῆς μᾶλλον τὸ εὖ ποιεῖν ἢ τὸ εὖ πάσχειν, καὶ τὰ καλὰ πράττειν μᾶλλον ἢ τὰ αἰσχρὰ μὴ πράττειν.
Eigen aan de deugd is eerder goed handelen dan goed ondergaan, en eerder het goede doen dan het slechte niet doen.
τῶν θεῶν ἐστι πάντα· φίλοι δὲ οἱ σοφοὶ τοῖς θεοῖς· κοινὰ δὲ τὰ τῶν φίλων. πάντ’ ἄρα ἐστὶ τῶν σοφῶν
All things belong to the gods. The wise are friends of the gods; and friends hold things in common; Therefore all things belong to the wise.
τί δ’ οὐκ ἐμὸν ἔσσεται αἶσχος;
Which disgrace will not be mine?