Heb 13:13 helps to understand the paraenetical purpose of the epistle. The term ὀνειδισμός and the appeal to go outside the camp (ἔξω τῆς παρεμβολῆς) highlight the problem of the intended reader as well as the author's pastoral strategy.
The writer demonstrates his concern for the readers living in non-Christian society where they experience suffering and some sort of shame. It is the Christ's form of being.
The author points the readers to the counter-world of "outside" marked by suffering as well as glorious victory of Jesus which establishes the identity of the Christian community. The popular hypothesis of recipient's desire to return to Judaism can't be satisfactorily proved out of Heb 13.
This article wants to contribute to the discussion on the social and pastoral profile of the recipients as well as the relationship between exposition and exhortation in the letter. This examination of Heb 13:13 may also bring some impulses to think of the Christian-minority identity in many countries of the world.