What is the meaning of "authentic according to Muslim's criteria" or "according to the criteria of the Two Sheikhs" in hadith books?

The saying "Authentic according to the condition of al-Bukhari or Muslim or the Two Shaykhs" carries two possible meanings: The first is that the narrators of the hadith are the narrators of al-Bukhari or Muslim, and this is the more accurate meaning. The second is that the narrators of the hadith are similar to the men of al-Bukhari or Muslim in terms of justice ('adalah) and precision (dabt). Al-Bukhari stipulates the fulfillment of actual meeting and narration between the narrator and the one from whom he narrated, whereas Muslim suffices with contemporaneity and the possibility of meeting.