This is a question concerning the proposition 11.4.8 of P.Hirschhorn's book Model categories and their localizations.
The proposition 11.4.8 is an analogous, in my opinion, to the well known statement in the topological spaces case : every cell of a cell complex is contained in a finite subcomplex of the cell complex.
Let me state the proposition first:
Let $\mathcal M$ be a cofibrantly generated model category in which cofibrations are monomorphisms. If $\gamma$ is a cardinal and $K$ is a set of cofibrations whose domains are $\gamma-$compact, then every $K-$cell of a relative $K-$cell complex is contained in a sub relative $K-$cell complex of size at most $\gamma$.
PROOF: Let $f:X\rightarrow Y$ be a relative $K-$cell complex, then we can write $f$ as the transfinite composition of a $\lambda-$sequence $X=X_0\rightarrow X_1\rightarrow X_1\rightarrow X_2\rightarrow \cdots\rightarrow X_{\beta}\rightarrow\cdots(\beta<\lambda)$ in which each map $X_{\beta}\rightarrow X_{\beta+1}$ is a pushout of an element of $K$. We will show by induction on $\beta$ that the attaching map of each $K-$cell factors through a sub relative $K-$cell complex of size at most $\gamma$. The induction is begun because the attaching map of the $K-$cell of presentation ordinal $1$ has codomain $X=X_0$.
We now assume that $\alpha < \gamma $ and that every $K-$cell of $X\rightarrow X_{\alpha}$ is contained in a subcomplex of size at most $\gamma$. Let $C\rightarrow D$ be an element of $K$ such that $X_{\alpha+1}$ is constructed as the pushout $$\begin{array} x C&{\longrightarrow}&D\\ \downarrow{h^{\alpha}}&& \downarrow \\ X_{\alpha}&{\longrightarrow}&X_{\alpha+1} \end{array} $$ we must show that $h^{\alpha} $ factors through a sub relative $K-$cell complex of size at most $\gamma$. By a previous lemma, we can find a diagram $$\begin{array} AX_0 &\stackrel{\sigma_0}\longrightarrow & X_1&\stackrel{\sigma_1}{\longrightarrow} & X_2&\stackrel{\sigma_2}{\longrightarrow}&\cdots& \longrightarrow &X_{\alpha} \\ \downarrow{i_0}&& \downarrow{i_1}&&\downarrow{i_2}&&&&\downarrow{i_{\alpha}} \\ \tilde{X_0} &\stackrel{\tau_0}\longrightarrow & \tilde{X_1}&\stackrel{\tau_1}{\longrightarrow} & \tilde{X_2}&\stackrel{\tau_2}{\longrightarrow}&\cdots &\longrightarrow &\tilde{X_{\alpha}} \\ \downarrow{r_0}&& \downarrow{r_1}&&\downarrow{r_2}&&&&\downarrow{r_{\alpha}} \\ X_0 &\stackrel{\sigma_0}\longrightarrow & X_1&\stackrel{\sigma_1}{\longrightarrow} & X_2&\stackrel{\sigma_2}{\longrightarrow}&\cdots& \longrightarrow &X_{\alpha} \end{array}$$ such that $r_{\beta}i_\beta=1_{X_{\beta}}$ for $\beta\leqslant \alpha $ and every $\tau_{\beta}$ is a relative $I-$cell complex (As usual, $I$ is the set of generating cofibrations). Thus the composition $\tilde{X_0}\rightarrow \tilde{X_\alpha}$ is a relative $I-$cell complex, and so the composition $i_{\alpha}h^{\alpha}:C\rightarrow \tilde{X_{\alpha}}$ must factor through some sub relative $I-$cell complex $\tilde{X_0}\rightarrow V$ of $\tilde{X_0}\rightarrow \tilde{X_{\alpha}}$ of size at most $\gamma$. We will complete the proof by showing that the composition $V\rightarrow \tilde{X_{\alpha}}\rightarrow X_{\alpha}$ factors through a sub relative $K-$cell complex of $X_0\rightarrow X_{\alpha}$ of size at most $\gamma$.
For each $I-$cell of $V$ there is exactly one $\beta\leqslant \alpha$ such that that cell is a part of $\tau_{\beta}$, and (by the induction hypothesis) the corresponding relative $K-$cell $\sigma_{\beta}:X_{\beta}\rightarrow X_{\beta+1}$ is contained in a sub relative $K-$cell complex of $X_0\rightarrow X_{\alpha}$ of size at most $\gamma$. If we take the union $Z$ of these sub relative $K-$cell complexes, then the relative $K-$cell complex $X_0\rightarrow Z$ has size at most $\gamma$, and the composition $V\rightarrow \tilde{X_{\alpha}}\rightarrow X_{\alpha}$ factors through the inclusion $Z\rightarrow X_{\alpha}$.
My question is: how to prove the last sentence of the proof:
the composition $V\rightarrow \tilde{X_{\alpha}}\rightarrow X_{\alpha}$ factors through the inclusion $Z\rightarrow X_{\alpha}$.
The proof is a little bit long and we can just start from the third paragraph 'For each $I-$cell of $V$ there is exactly one ...'
PS: I want to apply this proposition to the case when the model category is compactly generated in the sense of Bertrand Toen (cf. Moduli of Objects In DG-categories Definition 2.1.4 ) : each $I-$cell of an $I-$cell complex is contained in a strict finite sub $I-$cell complex and hence every object is weak equivalent to a filtered colimit of strict finite $I-$cell complexes.
