How does the idea of ``feedback'' get reflected in $\hat{F}$? Is there analogy of the function of $\hat{F}$ to the thermal control example?

Well, the most general concept of feedback is as follows: you take information from the output of a system and feed it back into the input to adjust the output. If it's ``negative feedback'', an output value is used in the input to adjust the output negatively (reduce an increase, increase a reduction). If it's ``positive feedback'', the output value is used to adjust the output positively (make something positive more positive, or something negative more negative).

In the thermostat example, you might measure the temperature at the output; if it's higher than the target, you send a signal to the input of your thermal control unit to reduce the temperature; if the temperature is lower than the target, then you send a signal to the input to increase the temperature. The ``feedback box'' implementing $\hat{F}$ is then a somewhat complicated thing (maybe a thermistor, some logic to send a control voltage to the input or something), rather than just the simple voltage divider that we had in our example. But it's still performing an $\hat{F}$ function of converting an output into some control value for the input.