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This is true for orthogonality classes- see Corollary 6.24 in Adamek and Rosicky - but I can't seem to find this result in the literature for weak orthogonality.

Here, by a weak orthogonality class in a category $C$ I mean a class of morphisms $R \subseteq Mor(C)$ such that $R= llp(rlp(R))$, and by a small weak orthogonality class I mean that $R=llp(rlp(I))$ for some small set $I \subseteq Mor (C) $. Here, if $A \subseteq Mor ( C ) $ then $ llp (A) $ denotes the class of morphisms with the left lifting property with respect to $A$ and $ rlp (A) $ denotes the class of morphisms with the right lifting property with respect to $A$.

I expect the hypothesis of local presentability and the assumption of Vopenka's principle to be necessary, as in the case of strong orthogonality.

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    $\begingroup$ There are many weak orthogonality classes in locally presentable categories which are not small. For instance, embeddings in posets (see my joint paper with Adámek, Herrlich and Tholen in Appl. Cat. Str. 10 (2002), 237-249). $\endgroup$
    – Jiří Rosický
    Commented Feb 26, 2017 at 12:09
  • $\begingroup$ @JiříRosický Thanks, that definitely answers my question! Is it still true under Vopenka that every weak orthogonality class in a locally presentable category yields a weak factorization system? This almost follows from the fact that every injectivity class is weakly reflective under Vopenka. $\endgroup$
    – Tim Campion
    Commented Feb 27, 2017 at 5:39

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