$\newcommand{\Vect}{\mathit{Vect}}$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 $\mathrm{source} \times \mathit{target}$ 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.
EDIT : I am sorry I didn't explain it clearly enough. Please let me explain my thinking by using a toy model of vector spaces, from a set-theoretic foundationalistic perspective. (To highlight this view, I add some links to pages having set theoretic definitions.) Defining an isomorphism class, as is usually done, should mean the following manipulation: Let $\Vect$ be the set of all vector spaces (of course this is the proper class though, I'll leave the details out.) we define the sets $\sim$ as $\{ (V_1, V_2): V_1 \cong V_2\}$. Trivially since $\sim \subset \Vect \times \Vect$ this is the (binary) relation on $\Vect$ (here I adopted the set-theoretic definition as there is in the link), and also, equivalence relation as well (one can prove this by using properties of isomorphism.) Therefore, one can define the isomorphism class of a vector space $V$ by $\sim$ as $[V]:= \{V' \in \Vect: V' \sim V \}$. On the other hand, however, if we replace $\sim$ with a poly isomorphism $\Xi$, then how can we define an isomorphism class? The first thing we need to do is, to define a relation $\{ (V_1, V_2): \phi(V_1, V_2) \}$. That is, we need to find a formula $\phi(V_1, V_2)$ using the set of isomorphisms $\Xi$. It seems to me that this is not obvious. For example, if we consider $\phi(V_1, V_2)$ as "$|\Xi| > 0$", then this is (from the definition of isomorphy between two objects) the same as the above relation so meaningless. Hence, the definition of an isomorphism class (in other words an object determined up to isomorphism) using $\Xi$ does not make sense (I think this cannot be solved even if one shifts the mathematical foundation from ZFC to another one.) I think Mochizuki's definition of ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ is under the same circumstance. Are there any mistakes in my thinking? If so, I would appreciate it if you could tell me.
