
Quoting
eapls2708
I agree with what Mr. Knowitall said with the possible exception of nailing boards to the fence.
Your neighbor has the right to prevent your cat's access to his or her property. Since you describe it as being on your deck, I'm picturing that you are in a duplex, a townhouse, or in a subdivision where many of the homes and improvements, including your deck, are built at a zero setback (essentially on the property boundary). If your lattice fence is on or very close to the boundary line, it would likely be considered a boundary fence, very much like the wall within a duplex or townhouse that separates one residential unit from another under the same roof is a party wall. The hole in the fence was made specifically for that purpose, so it follows that the neighbor must specifically prevent access through that hole to keep your cat from entering.
If that's the case, then the neighbor could attach some manner of material to the fence to block that hole, but it must be done in a manner that leaves it in a safe condition. Alternatively, and perhaps before blocking the hole in the manner done, your neighbor should have spoken to you and asked you to close up the hole. Presumably you made the hole for your cat, and so it was actually your responsibility to prevent the trespass.
Either way, you've been put on notice that the permission for your cat to access your neighbor's yard has been rescinded or revoked. Ensuring that your cat does not enter that yard is a shared responsibility. Your neighbor has the responsibility to take reasonable steps to prevent people and animals from entering their yard, and you have the responsibility of taking reasonable steps to ensure that your neighbor's right to privacy and a cat poop free yard is respected in so much as you and your family and pets are concerned.