Thanks for that Popular Electronics schematic, had one of those "Aha!" moments because of it.

That was because there is no obvious multiplier (or frequency mixer) there, so how does it shifts the spectrum from ultrasounds to audio? C5 and C10 only adds the ultrasound signal \$f_x\$ with the local oscillator \$f_{LO}\$. An adder does not produce new frequencies. If we add two frequencies \$f_x\$ and \$f_{LO}\$ we do NOT get any \$f_x - f_{LO}\$ like we get from a normal mixer/multiplier.

A circuit needs some non-linearity to produce new frequencies, frequencies other than the ones that were put in. So, how come that an adder can produce \$f_x - f_{LO}\$? Well, it doesn't.

Then how that that schematic even works? It's the diode detector with D1, D2, R8 following after the C5 and C10.

The diodes detector is in fact a frequency mixer, didn't realized that before!

Those diodes also acts as a frequency mixer

, then R8, C6, C7, R9 makes a low-pass filter that rejects higher frequencies and let to pass mostly the \$f_x - f_{LO}\$ component, which is the bats' ultrasounds shifted down to audio.

Added a screen capture to illustrate a low pass of \$f_x + f_{LO}\$ in blue vs a low pass of \$|{f_x + f_{LO}}|\$ in yellow.

The green trace is the sum of the two frequencies and the red trace is the rectified sum. In blue is the signal after low-pass on the sum of the \$f_x\$ and \$f_{LO}\$, while the yellow trace shows the same but for the absolute value (with diodes). Blue signal only shows ultrasounds, while the yellow signal shows \$f_x - f_{LO}\$.