I suspect there is a strong connection between these two results, but I'm not able to work out the details; in particular I sense that there is a strong analogy between the SOA, stated in the following way
Let $\Sigma$ be a set of arrows in a cocomplete category $\bf E$; if the domains of the maps in $\Sigma$ are $\alpha$-compact[1] for a regular cardinal $\alpha$, then there exists a functor $$ R\colon \bf E\to E $$ together with a natural transformation $\varrho\colon 1\to R$ such that:
- The object $R(X)$ is $\Sigma$-fibrant[2] for every object $X$
- the map $\varrho_X$ belongs to $\text{Cell}_\alpha(\Sigma)$[3] for every $X\in \bf E$.
and the solution to the OSP, as it is stated in the famous work by Freyd and Kelly, Categories of continuous functors 1, which says, more or less,
under suitable "compactness" conditions on a complete and cocomplete category $\cal A$ and on a class of objects $\Delta$, the class of arrows orthogonal to (all the objects of) $\Delta$ form a reflective subcategory.
To my eye the first statement is a direct consequence of the second: the functor $R$ is no more than the idempotent monad $iR$ induced by the reflection $R\colon \mathbf E \to \Phi(\Sigma)\colon i$, and $\Sigma$-fibrant objects are defined precisely via an orthogonality condition.
Skimming through Garner's "Understanding the SOA" paper I'm not able to see a precise statement in this sense ([FK] paper is cited only at the very beginning). Maybe I'm wrong, or this interpretation of the SOA is implicitly understood? If I'm wrong in some subtle detail, can you tell me which one? If I'm not wrong, I'm interested in making this analogy precise! Can you give me references where this is stated properly?
Thank you for your attention.
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Let me just fix notations:
1] It means that for each $S\in\Sigma$ the functor $\mathbf E(S,-)$ commutes with $\alpha$-filtered colimits.
2] An object $X$ is "$\Sigma$-fibrant" iff its terminal arrow $X\to 1$ belongs to the class $\text{rlp}(\Sigma)$ of all morphisms having the right lifting property with respect to each arrow in $\Sigma$.
3] A class of arrows in a category is said to be cellular if it contains all isomorphisms, is closed under finite and $\alpha$-transfinite composition, and under cobase change. $\text{Cell}_\alpha$, the cellularization operator, is the closure operator generated by the property "being cellular".