Mine connect together on the sides with industrial sticky back velcro. I never put the "collar studs" on the sides of my codpiece, because that entire area is covered up by the horse girth belt anyway, and nobody ever sees it.
I've attached web strapping and clips on the cod and butt and just clip in place.....its a great and secure way, but it's not as easily adjustable as it could be so with my fluctuating size I'm thinking of changing to industrial strength velcro.
In post number 219 I show, how and where I put industrial velcro later to keep it from slipping down. Your question was "How to connect cod and kidney?" Therefore I used elasctic, like Fett 4 Real said.
I use a piece of velcro on the front of the flight suit pants to keep it up around the waist and use the screw and nut (imbedded in a keyboard key) to attach the cod to the kidney plate on the sides (like connecting the collar plate to the back plate.)
I drilled a couple holes through the cod piece on either side to fit little screws through that I use those acorn nuts on. Hot glued the screws onto the kidney piece. Easy to attach, too small to notice, and very solid.