What is the guy in green showing to the F/A-18E pilot before catapult launch?
That's a weight check. The sign has the weight of the aircraft in pounds. The catapult crew guy first shows the presumed weight to the PIC who must give a thumbs up, agreeing, "Yes, that is the weight I believe my aircraft to have."

The crew guy must then show the same exact set of numbers to the catapult chief operator, who must also approve it: "Yes, the catapult is set to launch that weight.

You can actually see both verifications happening in the two photographs above, the first one is to the PIC, the second to the catapult operator. Only when both verifications have occurred will a launch be permitted.