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To make sure your seat is flush against the window, you’ll need to measure and notch the board to account for any trim work. To keep the wood from splitting, always drill pilot holes before putting screws into the wood. Next, anchor you brackets on the outside of the window.
A board that’s 10-12" wide will let your cat stretch out comfortably on the seat. With your brackets mounted on the wall, more support can be added, if you need it, by stapling cloth strips to the inside edge, letting the tail hang 2-4". For a more secure fit, you can notch the window seat so it fits inside the frame and up flush against the window. When the window is shut, the seat is secure.