Maybe try buying a chicken coop. I foundnone on amazon for under $100. It was about 4 ft high and 8 feet long. Put a cat door in your lanai and push the door of coop right up to cat door and then listen e bricks or something around outside so it cant move so the cat can go in and out but is 100% enclosed. And its enclosed by metal not screen so it should be coyote proof. Also the cat can quickly run into the house if needed
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