Let $X$ be a 2-site and consider the category of 2-presheaves over $X$, which will be denoted as $Pshv(X;Cat)$. These are $Cat$-valued 2-functors, where $Cat$ is the 2-category of categories. There is a functor "2-Sheafification" $L:Pshv(X;Cat)\rightarrow Shv(X;Cat)$, which is left adjoint to the inclusion $i:Shv(X;Cat)\rightarrow Pshv(X;Cat)$.

My question is: How does 2-Sheafification work with 2-presheaves valued in 2-categories other than $Cat$? I have in mind 2-categories with "extra structure", for example 2-functors valued in the 2-category of monoidal categories.