Wow...learned something...imagine what the big ones can do. I'm glad they are not crocs!

How the heck did it get in the courtyard villa? The courtyards have a gate etc

I looked at the video and that little alligator climbed over a chain link fence. Those are not at all similar to the concrete walls of a villa. The alligator would have footholds in the chain link but not on the concrete.

Now, if the OP left his gate open, it is possible but the gator would also be free to walk back out the same way.

Alligators could climb up our seawalls where I lived in Southwest Florida. They were concrete slab walls. Thankfully they rarely came up in the canals, but it has happened.