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.