Adding water softener. Confused by check valves (photos and descriptions)

I want to add a water softener to the right of my tankless water heater. So the cold supply (softened) line will be coming from right to left. However there are two check valves in the correct setup (left to right); I am not totally sure their intended purpose. Whether they are to prevent hot water being back-fed from the tankless or something to do with the in floor heater.

I attached some sketches of the current setup, and two proposed setups.

My two proposed setup and two more photos of my best guess of where to put the check valves. In the last two photos of each setup the orange markup is where I guess the check valves go.

I think the first setup might be wrong as the second forced the water from the in floor heater to always return to the tankless water heater and not the main line.

Any advice is appreciated.