linux - How Ctrl-C propagates through tmux -

i know when hit ctrl-c, operating system pts system interprets , sends sigint foreground process group. wonder how propagates through tmux.

say running normal linux x server , window manager. opened xterm , in zsh running tmux. in tmux running bash , in ./a.out.

process tree window manager -> xterm -> zsh -> tmux client. tmux server -> bash -> ./a.out.

so when hit ctrl-c on keyboard, x server first catches , tells active window xterm "hey guy hit ctrl-c". xterm tells process it's directly running (zsh) hit ctrl-c sending pseudo terminal pts/0 "ctrl-c" character. operating system saw "ctrl-c" character in pseudo terminal , translates sigint foreground process group (which zsh , tmux client). tmux client must have received sigint tells tmux server , tmux server sends sigint foreground process group of pseudo terminal created (with bash , ./a.out in it). sorry if looks confusing. because of how tmux works, tmux server , tmux client 2 separate processes.

now if want mimic behavior, send sigint tmux client , expect ./a.out receive sigint if pressed "ctrl-c" on keyboard. nothing happens. guess wrong somewhere , can me it?