$\newcommand{\Vect}{\mathit{Vect}}$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][1], 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)][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][3], 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][4] 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][3] 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][5] on $\Vect$ (here I adopted the set-theoretic definition as there is in the link), and also, [equivalence relation][6] as well (one can prove this by using properties of isomorphism.) Therefore, one can define the [isomorphism class][7] 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.


  [1]: https://ncatlab.org/nlab/show/initial+%CE%98-data
  [2]: https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf
  [3]: https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf
  [4]: https://proofwiki.org/wiki/Isomorphism_is_Equivalence_Relation
  [5]: https://en.wikipedia.org/wiki/Binary_relation#Definition
  [6]: https://en.wikipedia.org/wiki/Equivalence_relation#Definition
  [7]: https://ncatlab.org/nlab/show/isomorphism+class