Mind is a Function or 'set of action', which is done by oneself one after the other. That is, after 'observation' of an object, Mind functions takes place or more correctly, one does the 'actions' one after the other starting from Sensation -> then Cognition -> then Memory -> finally Thinking (Perceiving the object).
Though the 'observation' is an action of the mind, it is not included in the Function of the Mind, where Mind Function is referred to as the Mind.
By practicing the 'Attention' on breathing, in Buddhist Bhavana (Mind Development), one gets the skill of the 'Attention', where 'Attention' itself an action, but not that of the function of the mind, which helps to get rid of the Functions or actions of the mind, one by one to have different stages of Samadi.
First the skill of the 'Attention' helps to get rid of distracting 'Thinking', where such thinking takes away the 'Attention' from the object, i.e. from the breathing. Practicing Attention helps fixing the 'thinking' on the breathing itself, i.e. in the Bhavana object itself. Once the thinking is fixed, it is First Jhanic Samadi.
In this First Jhanic Samadi, where one of the Function of the Mind is brought under control (not controlling), that is 'Thinking' is under control, and deviant thinkings are stopped.
By the time one comes to the First Jhanic Samadi, the 'Attention' becomes a 'Skill', meaning one no longer need to make and effort to maintain the 'Attention', but it becomes effortless, continuous and this state of Attention is called as 'Sati'.
In the Second Jhanic Samadi, 'Thinking' on the object of Bhavana, i.e. on breathing is stopped to Cognize the Breathing continually through the 'Sati'. So at this stage two functions of the mind are stopped. That is 'Thinking' and the 'Memory' which is needed to think. Here one pays 'Sati' to the Cognition of the Breathing.
In the Third Jhanic Samadi, Cognition of the 'object' is stopped and one pays 'Sati' to the Sensation of the object of the Bhavana, i.e. Breathing.
In the Forth Jhanic Samadi, one stops the Mind function of the Sensation on the Breathing, and hence, there is no detection of the breathing through 'Sati', but 'Sati' remain focused on the place, where the 'breathing' was. This is made further possible by breathing getting 'suspended' at this stage. So, at this Fourth Jhanic Samadi, the functions or actions of the mind is Stopped altogether. But other functions, such as 'observations' and 'Sati' remains.
At Forth Jhanic Samadi, without having Sensations, one is not attracted by pleasant Sensations or not dispelled by unpleasant Sensations. Hence, one is in Equanimity here, and steady, without falling to 'this or that'.