Measure the width of the window.
Cut 1/4-inch thick plywood to make two rectangles, each of which is 14 inches tall and as wide as your window.
Clamp the two rectangular pieces of plywood together so you can cut and drill them as one later.
Draw a rectangle that is 8 inches wide and 10 inches tall in the center of one of the plywood rectangles.
Drill a 1/4-inch hole in each corner of the drawn rectangle, and then cut out the 8-by-10-inch rectangle with a jigsaw. The resulting open space is the door hole.
Drill four 1/4-inch holes in a line above the door hole. Also drill one hole in each corner of the plywood, positioning each hole 1 inch inward from the plywood's edges.
Remove the clamps, and sand the plywood with 120-grit sandpaper to remove splinters.
Cut a piece of thick, flexible plastic so it measures 8 by 12 inches. Scissors can be used to cut the plastic.
Lay one of the plywood rectangles on a table, and lay the plastic on top of that plywood piece. Arrange the plastic so that its sides and bottom line up with the door hole, with the overlap extending past the top and over the drilled holes.
Mark the positions of the drilled holes on the plastic, and then drill out those marks on the plastic.
Place the two pieces of plywood and the plastic together like a sandwich, lining up all of the drilled holes.
Bolt everything together with eight 1/4-inch diameter bolts that are 1 inch long. Put nuts on the ends of the bolts.
Open the window, and place the cat door in the opening. Close the window onto the top of the cat door. The cat door's plastic flap swings back and forth, allowing your cat to pass freely.