I'm reading a few papers on reflective factorization systems and I've just noticed they're all mentioning a procedure which seems very similar to the small object argument.

First of all, some background. My first encounter with the small object argument was Garner's paper about which I've asked several questions and learned a lot from. Lately I've been revisiting the basic theory of strong factorization systems in the context of categorical Galois theory and its connection to semi-left-exact reflections. I do not have a good understanding of the transfinite subtleties of small object arguments, but as far as I understand, what they do is freely construct *cofibrantly* generated factorization systems (weak or strong).

This question is about an apparent similarity between the goal of the small object argument, and an important procedure proving the finite well-completeness implies that reflective *pre*factorization systems are actually factorization systems.

Let $F\dashv G$ be an adjunction between $\mathsf C\overset{F}\rightleftarrows \mathsf B$. The prefactorization system on $\mathsf C$ *fibrantly* generated by the image of $G$ is $(\mathcal L,\mathcal R)=(\operatorname{\!^\perp Im}G, (\operatorname{\!^\perp Im}G)^\perp)$. Generally speaking this is not a strong factorization system. Let $f$ be an arrow. Seemingly the only way to get any interesting factorization of it is to factor the naturality square below through the pullback of $B\rightarrow GFB\leftarrow GFA$. This yields a factorization $f=v\circ w$, where $v$ is the pullback of $GFf$ along $\eta_B$ and $w$ is the unique arrow induced by the universal property of the pullback $P$. By definition $GFf\in \mathcal R$, and since $\mathcal R$ is always universal, so is $v$. Hence, in a sense, we already have "half" the $(\mathcal L,\mathcal R)$-factorization of $f$.

$$\require{AMScd} \begin{CD} A @>{\eta_A}>> GFA\\ @V{f}VV @VV{GFf}V\\ B @>>{\eta_B}> GFB \end{CD}$$

$w$ though is not generally in $\mathcal L$, which is always the class of arrows inverted by $F$. However, since $\mathcal R$ is closed under composition, finding an $(\mathcal L,\mathcal R)$-factorization of $w$ itself would suffice. The tentative strategy now is to take the intersection $m$ of all (strong) subobjects of the pullback object $P$ that are also contained in $\mathcal R$. If this creature exists, its universal property implies it must factor $v$ as $v=mg$ for some $g$, while limit closedness implies it's also in $\mathcal R$. Some careful arguments then show $g\in \mathcal L$.

The condition on a finitely complete category asking for intersections of strong subobjects to exist is called *finite well-completeness*. Adámek and Rosicky's book mentions locally $\lambda$-presentable categories as those in which every objects is a directed colimit of $\lambda$-presentable objects of a certain set, which seems at the very least analogous to finite well-completeness. Since locally presentable categories are basically "ones which admit a small object argument", I'd like to understand *precisely* the relation of the picture I sketched above with small object arguments in general.

Also, what can be done in the situation described if instead of an adjunction one just starts with some class of arrows $\Sigma$?