Proposition 22.1.153943AA7119C5

Proof. It is a tautology that the maps $P_{\theta +\omega }[\bullet ]_{\mathcal{I}}$ in definition 22.1.14 are compatible with the transition maps. So the map (22.1.16) is well-defined. It is injective by the same argument as proposition 22.1.6. We argue the surjectivity.

By unfolding the definitions, an object in the target of (22.1.16) is an assignment: With each $\omega \in \textup{Käh}(X)$, we associate a family $(\Gamma ^{\omega ,\omega '})_{\omega '\in \textup{Käh}(X)}$ satisfying:

1. 5067A37E976E

   $\Gamma ^{\omega ,\omega '}\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega +\omega ')_{>0}$ for each $\omega ,\omega '\in \textup{Käh}(X)$;

2. CDB9ED278B54

   for each $\omega ,\omega ',\omega ''\in \textup{Käh}(X)$ satisfying $\omega ''\geq \omega '$, we have

   \[ P_{\theta +\omega +\omega ''}\left[\Gamma ^{\omega ,\omega '}\right]_{\mathcal{I}}=\Gamma ^{\omega ,\omega ''}; \]
3. 3EBB3E3A9CF6

   for each $\omega ,\omega ',\omega ''\in \textup{Käh}(X)$ satisfying $\omega \leq \omega '$, we have

   \[ P_{\theta +\omega '+\omega ''}\left[\Gamma ^{\omega ,\omega ''}\right]_{\mathcal{I}}=\Gamma ^{\omega ',\omega ''}. \]

The preimage of such an object is given by the family $(\Gamma ^{\theta +\omega })_{\omega \in \textup{Käh}(X)}$ given by

The fact that the image of $\Gamma $ is as expected follows directly from the definitions. $\square$