How to build the following circuit?
When input is rising from 0V to 1V, output drop from 0V to -1V.
when input is dropping from 1V to 0V, output rise from -1V to 0V

 

How to build the following circuit?
When input is rising from 0V to 1V, output drop from 0V to -1V.
when input is dropping from 1V to 0V, output rise from -1V to 0V

Your description sounds like an inverting operational amplifier. Do you want to build or simulate one using MOSFETs? If so, the building locks would be:

1. Differential amplifier
2. Current mirror
3. Class A gain stage
4. Level shifter
5. Output stage

Are you familiar with these circuits? If not, then you should probably research them individually. This might be a good place to start

https://www.allaboutcircuits.com/technical-articles/the-basic-mosfet-differential-pair/

 

How to build the following circuit?
When input is rising from 0V to 1V, output drop from 0V to -1V.
when input is dropping from 1V to 0V, output rise from -1V to 0V

Is this a signal inverter (an inverting amplifier) or a power inverter (to produce a negative supply from a positive supply)?