At the end when the
group is laughing at the entity, Leonard Nimoy (Spock) is struggling
to stifle laughing along with them.
When Mara tries to
convince Kang to listen to Kirk, there is a close-up shot with her
and there's a intercom panel over her left shoulder. But in every
previous and subsequent wide shot she's standing in front of that
big wire-mesh grid that takes up one wall of Engineering.
No one notices the
Klingons are beaming down a party (despite the fact we've seen in
other episodes the sensors can sense transporter activity) or if
they did, nobody bothers to warn Kirk.
At the end when the
entity left the Enterprise, it came out of the secondary hull,
however it left from the engineering dept. The Engineering dept. is
located in the back of the primary hull.