I figured out that Spout might have been switched off and that was indeed the case.
In any case, I never knew Spout was the means of sharing Arena into Wire. I assumed they were connected like the siblings they are!
I might be wrong, but I've found no documentation regarding this functionality, which is pretty important!
It's only by chance that I had Spout enabled when I first looked at the Texture In dropdown and Arena-Composition appeared there.
Having:
Arena-Composition (Spout)/(Syphon)
greyed out as a dropdown item when Arena is not present, or Spout enabled, would instantly fix this problem.