Cut off 1/2-inch from the tip of a construction adhesive tube with a utility knife. Slip the caulk gun's puncture wire inside the adhesive tube's hole and puncture the seal inside. Slip the open construction adhesive tube into the caulk gun.
Angle the tip of the caulk tube against the wall studs and squeeze the gun's handle to produce a 1/2-inch bead of construction adhesive. Slowly slide the caulk gun down the wall stud, leaving behind a 1/2-inch continuous bead of adhesive the length of the plywood piece.
Continue applying construction adhesive on each wall stud where the plywood will go. Avoid applying the adhesive in areas where the plywood will not make contact unless you plan on applying another sheet of plywood soon after as construction adhesive dries quickly.
Place a sheet of plywood against the wall studs that you prepped with construction adhesive. Push against the plywood firmly to disperse the adhesive between the back face of the plywood and the wall studs.
Drive in two nails with a hammer, leaving the nail heads slightly higher than the face of the plywood, near the upper corners of the plywood to hold the sheet in place while the adhesive dries. Pull the two nails out of the plywood with the hammer's claw once the adhesive is dry.