I am reading Mochizuki's INTER-UNIVERSAL TEICHMÜLLER THEORY I to III and I hardly understand this theory, but there is a thing particularly I bother. Given an initial $\Theta$-data, consider $\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}$ Hodge theaters $\mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ defined in IUtchI Definition 6.13(2) and a poly isomorphism $\Xi$ between them(i.e. a set of isomorphisms between two $\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}$ Hodge theaters). Following this, let $\{{}^{m} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}\}_m$ be an infinite sequence depicted as follows (actually this is a part of a log-theta lattice though, I simplify by thinking only column): \begin{array}{rrcl} \cdots \xrightarrow{\Xi} {}^{-1}\mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}\ \xrightarrow{\Xi} {}^{0} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}\ \xrightarrow{\Xi} {}^{1}\mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}\ \xrightarrow{\Xi} \cdots \end{array} In IUtchIII Proposition 3.10, he introduces ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ for the $\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}$ Hodge theater determined up to isomorphism. It seems that this isomorphism is $\Xi$ according to the context, but however, as far as I know, an object determined up to isomorphism should represent the isomorphic class. That is, since isomorphism is an equivalence relation it is supposed to mean the equivalence class by regarding the isomorphism as an equivalence relation. Especially since such isomorphism is a relation, it must be a subset of $source \times target$ of the isomorphism, but $\Xi$ is clearly not because it is a set of isomorphisms. Therefore I feel ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ can't be defined and Proposition 3.10 does not make sense. Is this a mistake that is based on a lack of my comprehension? If you could answer me, I would be very happy. Thanks in advance.