I don't think my Co-Ax or any other one is fragile. But the last thing I'd do is put any press under running hot water - even if because of the question: "But why do that?". There's never water near the press during the reloading process - so why would I introduce water into the game just to clean it? Isn't that a fair question?
I probably don't have the bestest of the cleaning regimes, but what I do has worked for my Co-Ax since I bought it back in the 1970's.
I take a small stiff brush to all the residue that builds up on non-machined surfaces with a vacuum nozzle beside where I'm working to suck it all up, rather than move the residue somewhere else on my bench or the floor.
Then a plain old shop paper towel to wipe up as much oil and grease as I can get at on the moving parts as I cycle the press back and forth. That gets the majority of the oil/grease/lub that has picked up crap during operation.
Finally, I use Kroil directed into the internal bearing surfaces where the rods run, machine joints, etc to wash out what wasn't carried out to be wiped away before that. I have used Kleen-Flo at times when I couldn't find any Kroil handy.
All that done, I use Ponsness-Warren's STOS to lubricate all mating machined surfaces that move, and a very, very fine wipe of whatever gun oil is handy to protect exposed machined surfaces that don't move.
There may well be better ways to do it, and maybe better cleaners and lubricants to use instead, but it's worked just fine for me for almost 50 years now. I will go out on a limb and say it's a lot better than introducing soap and water into the press moving machined parts and linkages as the method of cleaning. Rather not deal with water when there's no need for it to be there in the first place.