I think you have two errors in your picture (the text is fine).
My understanding from the text is that you have proven:
If Alice’s latent is a mediator then (Alice’s is a natural latent <-> (for all possible Bob’s where Bob’s latent is a mediator and Alice and Bob agree on observables, Bob’s latent determines Alice’s)
The image leaves out the bolded bits, namely it leaves out that
Alice’s latent is assumed to be a mediator
The theorem is about what Alice needs to do to have a guarantee for all Bob’s.
That is, the image says: Bob mediation + Alice and Bob agree on observables → (Alice’s is a natural latent <-> Bob’s determines Alice’s)
I didn’t put the epsilons, and it’d easier to read if you swap the two parts of the conclusion in my image so that it’s like yours. That is, it’d be easier if I said [A is natural] <-> [every B that mediates is translatable to A]
The image you want should look like this[1]:
I think you have two errors in your picture (the text is fine).
My understanding from the text is that you have proven:
If Alice’s latent is a mediator then (Alice’s is a natural latent <-> (for all possible Bob’s where Bob’s latent is a mediator and Alice and Bob agree on observables, Bob’s latent determines Alice’s)
The image leaves out the bolded bits, namely it leaves out that
Alice’s latent is assumed to be a mediator
The theorem is about what Alice needs to do to have a guarantee for all Bob’s.
That is, the image says: Bob mediation + Alice and Bob agree on observables → (Alice’s is a natural latent <-> Bob’s determines Alice’s)
I didn’t put the epsilons, and it’d easier to read if you swap the two parts of the conclusion in my image so that it’s like yours. That is, it’d be easier if I said [A is natural] <-> [every B that mediates is translatable to A]