[forotis]

Yes.

The Buddha said: "these three qualities — right view, right effort & right mindfulness — run & circle around [the factors of the path]"

If the mind is watching the breathing but without an attitude of relinquishment, non-attachment or non-craving, that is not right mindfulness

The be right mindfulness (samma sati), sampajanna (clear comprehension, ready wisdom) must also be functioning