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When I am connected from my home computer to a server, and then connect to a 2nd server via ssh on the 1st server, I realized I am able to authenticate instantly without the private key being on the 1st server, only my home computer.

I've not seen this behavior mentioned so it freaked me out a little and I went around checking that the sshd was configured properly on both servers and not just letting anyone in.

I just wanted to confirm this is known, proper behavior because I cannot seem to google it.

Or is it actually that because I've connected to each server previously that it somehow decoded/stored the private key and now has it for re-use. In which case I have no idea where it is storing it (centos).

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After much more Googling and poking around my client settings, I discovered the terminology is called "Agent Forwarding"

This page gives a good description with diagrams:

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