When Cooper broke down in front of Izzy, he lied at first about having a trust fund.
He didn't tell Izzy the truth until they had fallen in love and Izzy felt guilty about taking his trust fund.
We have to assume that Izzy owned the buisness outright, as he was leaving the FBI. But still had a job with them, as his only target was Cooper.
But Izzy never expected to find himself, falling in love with Cooper. If Cooper hadn't revealed the truth about the money, we would probably never known that Izzy was a part-time FBI agent.