coproc redirect to stdout

Suppose I have some process that when ready will output to stdout. Even though I want my script to run asynchronously with respect to that process, I still want the script to block and wait on that first line. Modelled bellow with sleep and echo is exactly the behaviour I need in working order:

coproc monitor {
  sleep 2
  echo "init"
  sleep 1
  echo "foo"
  echo "bar"
  echo "baz"
}

read -u ${monitor} line
echo started
exec 3<&${monitor}
cat <&3 &

sleep 2

The script starts and creates the coprocess, then it waits on that first line via read -u, and finally it attaches 3 to ${monitor} so that we can then use cat in yet another background process to pipe stuff from monitor to actual stdout. Thus we get:

# waits 2 seconds
started
# after 1 second:
foo
bar
baz

I am not too happy with these two lines though

exec 3<&${monitor}
cat <&3 &

Is there no better way of achieving this? It seems like a rather roundabout of doing things. But everything I tried hasn't worked. For example cat <&"${monitor}" works, but then it blocks the script. { cat <&"${monitor}" ; } & gives Bad file descriptor for reasons I don't understand (but even then, I am still suing yet another background process, which seems sily).