The adapter is something he made himself. What it does is to keep a pin at the rear of the lens held in; this pin closes down the aperture to the chosen f/stop. With an M42-mount camera, the camera would have a mechanism to depress this pin. The K-mount cameras of course do not have such a mechanism, so there is nothing to hold the pin in, which means that you cannot stop down the aperture and therefore can only use the lens wide-open... unless you modify the lens as Mr Carpenter did.
This is not an issue with many other M42 lenses which have an Auto/Manual switch on the lens that allows you to close down the aperture on the lens instead of having to rely on the camera to do it. If you look at the photo of the two lenses side by side, you will notice that the Asahi has such a switch, just below the aperture ring.
P.s. if you read the text below the image, the Fujinon is on the "other left" and the Asahi is on the "other right"