$\require{AMScd}$If $\mathcal{X}$ is a category and $I$ a small category, the category of functors $\mathcal{X}^I$ inherits a (orthogonal) factorization system for each (orthogonal) factorization system on $\mathcal{X}$, defining the two classes objectwise.


It seems to me that I can define this factorization system "formally" in this way. Call $(\cal A,B)$ the factorization system on $\mathcal X$. Then there is a pullback diagram
$$
\begin{CD}
\mathcal{A}^I @>>> \mathcal{A}^{|I|} \\
@VVV @VVp\_\*V\\\
\mathcal{X}^I @>>j^\*> \mathcal{X}^{|I|}
\end{CD}
$$
where $j : |I|\hookrightarrow I$ is the inclusion of the discrete subcategory of $I$ into $I$ itself, $j^\*$ is the induced functor, and $\mathcal{A}$ is regarded a the nonfull subcategory of $\mathcal{X}$ on the arrows of $\cal A$ and $p\_\*$ comes from the obvious functor $\mathcal A \to \mathcal X$.


Is this rewriting correct?



Induced factorization system as a pullback in $\bf Cat$
If by $\mathcal{A}^I$ you mean the non-full subcategory of $\mathcal{X}^I$ corresponding to the left class of the induced factorization system (which is not the functor category of $I$ into $\mathcal{A}$), then yes, it does fit into such a pullback square. This doesn't construct the whole factorization system however.


