About 70C is the threshold of too hot to touch. Trannies have quite a range of temperature operation. Sometimes as they are often bolted to the engine, the engine temp will transfer to the tranny, so they'll run nearly as hot as the engine housing. They are designed to cope with a fairly wide operating range, so shouldn’t be cause for concern. Somewhere between 50C and 80C would probably be “normal”. I can ask colleagues for more accurate data if it's really of interest.

An AT would probably run hotter than an MT, especially is it doesn't have a lock-up toque converter clutch. If it gets so hot that water boils right off it (i.e. well over 100C) that's too hot. If it's too hot, things will fail faster. It will stress the gears and bearings, seals will have shorter life, friction plates will wear faster, oil might oxidise and gum-up working parts, which in a valve body of an auto box is bad. Less is likely to happen to an MT, but still better cooler.

As for if it’s cooler / too cool; they're designed to start from cold, so mostly they should function fine. As it warms to “normal”, shifting might be bit smoother (in an MT because the oil flow between synchros will be better and in an AT because flow to the clutch plates, around the torque converter and through pump and valve bodies will work a bit better. But if it's smooth on cold starts, “too cool” wouldn’t be an issue. I suppose his fuel economy might suffer a touch, but not that significantly. The only thing would be if this bloke is from the arctic circle and has extremely cold running conditions, in which case he'd need it warm enough to ensure good fluid flow around his box, but for anyone in the UK, that'll never be an issue.