PomeDefinition 22.1.2.657951EB240B We define
\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )=\varprojlim _{\omega \in \textup{Käh}(X)} \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0}{}^{\href{#sgpt-a0000106603}{1}} \]
in the category of sets, where the transition maps are given as follows: Suppose that $\omega ,\omega '\in \textup{Käh}$ and $\omega \geq \omega '$. Then the transition map is defined in proposition 17.14.6:
A99876729E5F
22.1.3
\begin{equation} P_{\theta +\omega }[\bullet ]_{\mathcal{I}}\colon \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega ')_{>0}\rightarrow \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0}. \end{equation}
Note that the transition maps are compatible:
\[ P_{\theta +\omega ''}[P_{\theta +\omega '}[\varphi ]_\mathcal {I}]_\mathcal {I}=P_{\theta +\omega ''}[\varphi ]_\mathcal {I},\quad \forall \, \varphi \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\omega )_{>0} \]
for $\omega \leq \omega '\leq \omega ''$, since both sides are $\mathcal{I}$-model and are $\mathcal{I}$-equivalent to $\varphi $.
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