ssh-agent for nested tmux session

Question

sessionA
    sessionB

I start tmux in first window of sessionA but using this command to wake it: ssh-agent tmux that I expect new nested sessionB will have its own ssh-agent listen for it. And output of ps aux | grep agent will show something like:

...    0:00 ssh-agent tmux

But instead I pass some options to tmux like: ssh-agent tmux <...> and expect ssh-agent still stay in-process for my new nested session, turn out it does but quickly ended/exited and the output of ps aux | grep agent doesn't show ssh-agent tmux <...> there.

So why ssh-agent tmux <...> get ended?

Answer 1

Complete answer:

If I put tmux <...> in subcommand I can keep ssh-agent tmux appears in output of ps aux | grep agent but now ssh-agent tmux is running in subshell that wouldn't terminate after server is killed.

EDIT for complete answer: I passed -d to detach when pass options to tmux so that killed ssh-agent process.