I think you might have misread something? The graphical statement of theorem 2 does not say that if ΛA is determined by ΛB, then ΛA is a mediator; that would indeed be false in general. It says that:
If ΛB is a mediator and we have agreement on observables, then...
… naturality of ΛA implies that ΛA is determined by ΛB.
In particular, the theorem says that under some conditions ΛA is determined by ΛB. Determination is in the conclusion, not the premises. On the flip side, ΛA being a mediator is in the premises, not the conclusion.
This was all clear to me, but only from reading the text; my comment is just to say that the graphical statement doesn’t show ΛA being a mediator in the premises, so in isolation it gives the wrong idea; this led to a little confusion.
To be clear, I am talking about the reverse direction, as pictured here:
I understand that you have already set up ΛA as a mediator immediately above the image. Your text is perfectly clear:
In other words, we want to show: if Alice’ latent ΛA satisfies Mediation, and for any latent ΛB Bob could choose (i.e. any other mediator) we have ΛA←ΛB→ΛA, then Alice’ latent must be natural.
The other problem is that the image has only a single B, but the actual theorem proves necessity of Alice’s being a redund from the requirement that hers is determined by all possible Bob’s (that are mediators and agree on observables). Without the for all, you can’t sub in X_1 and X_2 for his latent.
I think you might have misread something? The graphical statement of theorem 2 does not say that if ΛA is determined by ΛB, then ΛA is a mediator; that would indeed be false in general. It says that:
If ΛB is a mediator and we have agreement on observables, then...
… naturality of ΛA implies that ΛA is determined by ΛB.
In particular, the theorem says that under some conditions ΛA is determined by ΛB. Determination is in the conclusion, not the premises. On the flip side, ΛA being a mediator is in the premises, not the conclusion.
This was all clear to me, but only from reading the text; my comment is just to say that the graphical statement doesn’t show ΛA being a mediator in the premises, so in isolation it gives the wrong idea; this led to a little confusion.
To be clear, I am talking about the reverse direction, as pictured here:
I understand that you have already set up ΛA as a mediator immediately above the image. Your text is perfectly clear:
The other problem is that the image has only a single B, but the actual theorem proves necessity of Alice’s being a redund from the requirement that hers is determined by all possible Bob’s (that are mediators and agree on observables). Without the for all, you can’t sub in X_1 and X_2 for his latent.