The “only if” part follows from the following facts/observations.

• -If \(C\) is a \(U\)-presentable lex \(U\)-cocomplete category, then it is a lex localization of the category of \(U\)-small presheaves on a \(U\)-small lex category.
• -A lex localization of a \(U\)-presentable lex \(U\)-cocomplete category is a coinverter in \(\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\).
• -The category of \(U\)-small presheaves on a \(U\)-small lex category is \(U\)-compact in \(\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\).

For the “if” part, observe that the \(U\)-compact objects in \(\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\) are constructible under \(U\)-small colimits from the free lex \(U\)-cocomplete category over one object. Then the proof proceeds by induction.

• -The free lex \(U\)-cocomplete category over one object is the category of \(U\)-small presheaves on the free lex category over one object. Thus, it is \(U\)-presentable.
• -\(U\)-presentable lex \(U\)-cocomplete categories are closed under \(U\)-small colimits in \(\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\). This is proved by tracking the construction of colimits of \(U\)-presentable lex \(U\)-cocomplete categories given in Section 6.3.4 of [Lurie--2009-0000]. The key observation is that colimits of \(U\)-presentable lex \(U\)-cocomplete categories are decomposed as colimits of \(U\)-small lex categories and lex localizations both of which are colimits in \(\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\).