Commander Kelby took over as chief engineer on the NX-01 when Trip transferred to Columbia. Over the course of the episodes, it's established Kelby was newly promoted to the position and though he is meant to be a competent engineer, he is still somewhat raw at running the department. Later, when Trip transfers back to the NX-01 permanently, Kelby remains on the ship and even still keeps his rank despite the fact Trip is running the department again.
So what is the deal that he stayed on board the ship? Since Columbia was without a chief engineer, shouldn't he have transferred over there, basically trading places with Trip.
So what is the deal that he stayed on board the ship? Since Columbia was without a chief engineer, shouldn't he have transferred over there, basically trading places with Trip.