Let $\mathcal{C}$ be a cofibrantly generated model category, which is generated by $I$ and $J$. According to the small object argument (Hovey Theorem 2.1.14) of cofibrantly generated model categories, each map $X\to Y$ in $\mathcal{C}$ could be factored as $$X\xrightarrow{i} Z \xrightarrow{r} Y,$$ where $i$ is a relative $I$-cell complex and $r$ is a weak equivalence. But now, I need the cofibrantly generated model category $\mathcal{C}$ to satisfy the extra following property:

For any finite $I$-cell complex $X$ in $\mathcal{C}$, the fold map $(\mathsf{id}_X,\mathsf{id}_X)\colon X\amalg X \to X$ could always be factored as $$X\amalg X \xrightarrow{(i_0,i_1)} X' \xrightarrow{r} X$$ where $(i_0,i_1)$ is a relative finite $I$-cell complex and $r$ is a weak equivalence.

(To avoid misunderstanding, a relative finite $ I $‑cell complex means that both the factorization and the coproduct of morphisms in $I$ are finite.)

I think this property is not established for every cofibrantly generated model categories, but I also think it is not a very strong restriction. So I was wondering if there are some "deeper meaning" of this property (in a more general view)? or some large classes of cofibrantly generated model categories (maybe with extra structure such as enrich structure) has this property?

Additionally, the category $\mathbf{Top}$ (or $\mathbf{Sset}$) is too special for an example of this property, since the "middle object" $X'=X\times [0,1]$ of such factorization is generated by the catesian product, and finiteness of $X\times [0,1]$ also rely on the catesian product. And I also read Williamson's thesis Cylindrical model structures , it looks like it satisfies this property, but as I say I think this property may not depend on the "cylindrical structure".