Yes, and as you said, the work was directly job related, he developed a tool to make his work more efficient. Then he didn't tell about the tool to his employer. Obviously he should have.
They pay for his time. He should have disclosed that he can now perform the work much faster so that they can use his time for other stuff.
It's not like he developed a cool iPhone game in his spare time, unrelated to his job. The work was directly job related.