Lakay has penned a pre-contract with the Clever Boys effective July 1, but has since made the switch to Milpark on a short-term deal that will see Mxolisi Macuphu move the other way on loan.
Macuphu joined Wits at the start of the season from Chippa United and has managed two goals in 14 appearances thus far.
The 29-year-old has since fallen out of favour, having managed just 45 minutes of action in the club's last eight fixtures since the end of November.
Lakay, meanwhile, has also experienced sporadic game-time at Matsatsantsa since his return from a loan spell at Ajax Cape Town last season.
The exchange therefore means both clubs benefit, as do the players who will now be afforded fresh opportunities for more regular game-time.
Lakay could make his debut when Wits host relegation-threatened Maritzburg United on Friday evening, while Macuphu will have to wait until February 10 before the Tshwane side visit Golden Arrows.