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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.

$\mathbf{Edit}$: The following is from the nLab's entry on sheafification:

If a category A satisfies the following assumptions, sheafification of presheaves in $[S^{op},A]$ exists and is constructed analogously as for Set-valued sheaves.

A admits small limits;

A admits small colimits;

Small filtered colimits in A are exact;

A satisfies the IPC-property.

Maybe the answer to my question is related to the target 2-category satisfying 2-categorical versions of the above conditions?

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.

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.

$\mathbf{Edit}$: The following is from the nLab's entry on sheafification:

If a category A satisfies the following assumptions, sheafification of presheaves in $[S^{op},A]$ exists and is constructed analogously as for Set-valued sheaves.

A admits small limits;

A admits small colimits;

Small filtered colimits in A are exact;

A satisfies the IPC-property.

Maybe the answer to my question is related to the target 2-category satisfying 2-categorical versions of the above conditions?

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user84563
user84563
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"2-Sheafification" with Values in non $Cat$ categories?

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.