Stirling-type cooler (Air liquefaction)

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Hello. I'm currently studying air liquefaction processes and I've come to Stirling-type cooler.
With Linde, Claude, Heylandt and Kapitza cycles I find it very intuitive to get air cooled with throttling / expansion processes since they lower the temperature of incoming gas, but I am failing to understand how does a Stirling engine produce low temperature.
I understand the basics of Stirling cycle and It's operation between Thigh and Tlow, but at the start of the operation (let's say from the moment of turning on the engine), how does a Stirling engine achieve It's low temperature?

Is it right to conclude that both "air liquefaction chamber" and surroundings are at same temperature at the start of operation and Stirling engine keeps on "taking heat" from the "air liquefaction chamber" and dumping it to the surroundings until the temperature difference occurs?
 
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Mech_Engineer said:
Generally speaking a Stirling engine is similar to a refrigeration cycle. Heat is transferred from a "hot side" to a "cold side" using a working fluid and pressurizing the working fluid.

I find this article to be pretty well laid out regarding operation of a Stirling engine: https://en.wikipedia.org/wiki/Stirling_engine

Let's say that with regular "power-producing" Stirling engine, process starts with putting a candle on one side and leaving the other side open to the room temperature. Temperature difference drives the engine.

But I was wondering how does a reversed Stirling engine achieve low temperature? At the beginning of the operation both sides are open to the room temperature and continuous transfer of the heat from one side to the other side generates a temperature difference after some time. Is this correct?