Well, I found out something interesting recently. I commented to a Taiwanese person I know "Why is there a small WEST gate on the campus? Cheng Da's campus is way in the eastern part of old Tainan! Was it moved to preserve it or something?" (Actually, the district the campus is in is even called "East District", and there's a "Big East Gate" just south of campus, so it definitely is strange that Small West would be here.)
Well my friend told me the next time she saw me that she had looked it up, and she found out that the gate had been moved, just as I had guessed. Furthermore, this location is actually the original location of a gate. Apparently the "Small East Gate" was located here (and I believe the ruined walls around the gate are actually remains from the old city walls). So this location did originally have a gate, just not this one.
The top photo shows the front of the gate with its name, and the bottom two photos are showing the other side where its alternate name is. All of these gates had alternate names that were based on the significance of their compass position in Feng Shui. Small West Gate has "Clean/Pure Wave Gate" on its backside. I don't know the significance of this name though because I know very little about Feng Shui. Hopefully, having the Pure Wave Gate in the eastern part of town isn't giving us bad mojo now.