to oversee/care for
episkopeo denotes active oversight or watchful care, whether for God’s flock or to prevent failure and trouble.
The two verses show active, ongoing watchful care: in 1 Peter 5:2 it concerns shepherding God’s flock, while in Hebrews 12:15 it concerns guarding against anyone’s failure to obtain grace and the spread of trouble.
Senses in use
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Active oversight and watchful care · dominant
Both occurrences use the present active nominative plural masculine participle. In 1 Peter 5:2 the APB renders the clause “Shepherd the flock of God that is among you, overseeing,” describing care exercised willingly and eagerly; in Hebrews 12:15 it renders “See to it that no one fails to obtain the grace of God,” followed by concern that bitterness not cause trouble and defile many.
In the Anselm Project Bible: The APB presents the participle as “overseeing” in 1 Peter 5:2 and as “See to it” in Hebrews 12:15. These phrases confirm the clauses’ watchful-care context rather than establish separate meanings.