And of course Captain Christopher's jet falls apart and they have to beam him aboard the Enterprise. Kirk is generally amused by this and gives Captain Christopher a mini tour of the ship.
Y'know, until Spock tells Kirk that bringing Captain Christopher aboard the ship could mean the end of their future as they knew it and then Kirk has this look on his face like "maybe beaming him aboard isn't as funny as I thought it was!" Spock says that to amend the problem Captain Christopher can not return to Earth.
To add on to problems Scotty tells everyone that their little time jump has screwed up the warp engines and they're stuck in the 1960s. AND THEN...to make matters worse Captain Christopher says that he sent some pictures of the Enterprise to the Air force before he was beamed up. AND THEN... Spock does some scans in history and learns that Captain Christopher does have to go back down to Earth. His not yet born son will lead an expedition to Saturn. Problem is, the jet was totaled and the air force is snooping around the crash site so getting him down again really will look like aliens. AND THEN...when Sulu and Kirk go down to get the missing pictures they are caught by an Air Police Sergeant who makes them remove their belts, communicators, and phasers. When McCoy thinks they are taking too long Spock transports the communicator that emits an emergency signal up to the Enterprise. That communicator was in the hands of the Air Police Sergeant. AND THEN...when Sulu is figuring out the photos Kirk is caught by Air force Colonel Fellini. He almost gets Sulu, but Sulu beams up to the Enterprise with the pictures.
Spock, Captain Christopher, and Sulu then beam down to rescue Kirk which results in lots of punching. Back on the Enterprise Spock and Scotty figure out if they slingshot around the sun and escape its gravitational pull and all this other wonky stuff in just the correct amount of time they can get Captain Christopher and the Air Police Sergeant back to their times without them remembering anything that happened and the Enterprise back to its correct time too.
The maneuver's risky but they manage to pull it off anyway, getting everyone back to their correct time without a recollection of the Enterprise. And then they do the rest of the wonky maneuver and the Enterprise is back in their space and time. The Star Fleet wants to know whats up with the Enterprise and where they are, which Kirk responds, "We're home."