Subcomplexes of relative I-cell complexes

Question

This is a problem concerning the definitions given by Hirschhorn in his book Model Categories and Their Localizations 10.6.7.

Let $C $ be a cocomplete category and let $I $ be a set of maps in C. If $$(f:X\rightarrow Y,X={X_0}\rightarrow X_1\rightarrow X_2\rightarrow \cdots\rightarrow X_{\beta}\rightarrow \cdots ,(\beta<\lambda), \{T^{\beta}, e^{\beta},h^{\beta}\}_{\beta<\lambda})$$ is a presented relative I-cell complex, then a sub complex of $f$ consists of a presented relative I-cell complex $(\overline f:X\rightarrow \overline Y,X=\overline {X_0}\rightarrow \overline X_1\rightarrow \overline X_2\rightarrow \cdots\rightarrow \overline X_{\beta}\rightarrow \cdots(\beta<\lambda), \{\overline T^{\beta}, \overline e^{\beta},\overline h^{\beta}\}_{\beta<\lambda})$ such that

1. For every $\beta<\lambda$ the set $\overline T^\beta$ is a subset of $T^\beta$ and $\overline e^\beta$ is the restriction of $e^\beta$ to $\overline T^\beta$, and

2. There is a map of $\lambda-$sequences (The diagram is really not what I want: the arrows are not positioned correctly)$$\begin{align} &X\rightarrow &\overline X_0\rightarrow &\overline X_1\rightarrow &\overline X_2\rightarrow \cdots\\ &\downarrow & \downarrow&\downarrow&\downarrow\\ &X\rightarrow &X_0\rightarrow &X_1\rightarrow & X_2\rightarrow \cdots \end{align}$$ such that, for every $\beta<\lambda$ and every $i\in\overline T^\beta$, the map $\overline h^\beta: C_i\rightarrow \overline X_\beta$ is a factorization of the map $h^\beta_i:C_i\rightarrow X_\beta$ through the map $\overline X_\beta\rightarrow X_\beta$.

My question is, are the maps $\overline X_\beta\rightarrow X_\beta$ a relative I-cell complex, as is used in the proof of prop 10.6.10 of Hirschhorn?

Notations: for each $\beta<\lambda$ ,$T^\beta$ is a set and $e^\beta$ is a function: $T^\beta\rightarrow I$. For each $i\in T^\beta$ and $e^\beta_i$ is the element $C_i\rightarrow D_i$ of $I$,then $h^\beta_i$ is a map $C_i\rightarrow X_\beta$ such that there is a pushout diagram

$\begin{align}&\amalg_{T^\beta}C_i\rightarrow &\amalg_{T^\beta}D_i \\ &\downarrow &\downarrow \\ &X_\beta\rightarrow &X_{\beta+1} \end{align}$

Answer 1

At least this is true if we slightly change the definition: In 2. we require not only $\overline h^\beta: C_i\rightarrow \overline X_\beta$ is a factorization of the map $h^\beta_i:C_i\rightarrow X_\beta$ through the map $\overline X_\beta\rightarrow X_\beta$, but also the obvious map $D_i\rightarrow \overline X_{\beta+1}$ is a factorization of the map $\overline X_{\beta+1}\rightarrow X_{\beta+1}$. Then for each $\beta$, the map $\overline X_\beta\rightarrow X_\beta$ is induced by the pushout and is unique. Now just observe that $\overline X_\beta\rightarrow X_\beta$ is the pushout of the coproduct of maps in $T^{\beta}- \overline T^{\beta}$.

I am not sure if this remains true when the definition is not changed.

I am so sorry for the use of not mentioned notations.