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 $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$.