Police Academy/WMG

Lassard knows full well that Harris hates him...
...and he takes great pleasure in making the Jerkass' life as miserable as possible. That's why he's always so happy to see him.

Tackleberry was discharged from the military under Section 8.
When the police academy drops its standards, Tackleberry is taken on without question.

Zed was allowed to join the police academy because he never killed or seriously injured a person...
As the crazy punk he was, he was more interested in just scaring them and doing what ever he liked. Despite his violent behavior he never committed harsher crimes and during his knife fight with Mahoney he only toyed with him. He also pleaded guilty in court and was released early for good behavior. The reason he shows up at the academy in 3 is someone gave him the same chance as Mahoney to proof himself as willing to change.
 * Maybe it was Captain Reed.

Harris and Mouser worked together before.
Due to their similar personalities (and their hatred of Mahoney and the misfits). Mauser may have been the one who assigned Proctor to Harris sometime between 3 and 4. And Harris had Blankes and Copeland worked for Mauser in 3.