Embedding category-valued functors into indexed categories Let $\mathfrak{cat}$ denote the $1$-category of locally small categories, and $\mathfrak{Cat}$ denote the $2$-category of locally small categories. Further, for a category $\mathcal{C}$ let $\mathcal{C}^\uparrow$ denote the discrete $2$-category promotion of $\mathcal{C}$. We trivially have a $2$-embedding $$[\mathcal{C},\mathfrak{cat}]^\uparrow\hookrightarrow[\mathcal{C}^\uparrow,\mathfrak{Cat}]$$ from the discrete promotion of the functor category from $\mathcal{C}$ into $\mathfrak{cat}$ to the pseudofunctor category from $\mathcal{C}^\uparrow$ to $\mathfrak{Cat}$, with the latter being the '$2$-category of $\mathcal{C}$-indexed categories'. Staying low dimensional, my question is

What nice properties does this embedding have?

To start with, this should be related to the right adjoint of the truncation functor from $2$-categories to $1$-categories and should thusly preserve limits.
Moving up the dimensional hierarchy, we can repeat this process for an $n$-category $\mathcal{C}$ by letting $\mathfrak{cat}_n$ denote the $n$-category of $n$-locally small $n$-categories and $\mathfrak{Cat}_n$ denote the $n+1$-category of $n$-locally small $n$-categories, and $\mathcal{C}^\uparrow$ denote the discrete $n+1$-category promotion of an $n$-category $\mathcal{C}$. We trivially have $$[\mathcal{C},\mathfrak{cat}_n]^\uparrow\hookrightarrow[\mathcal{C}^\uparrow,\mathfrak{Cat}_n]$$ once again, and I'm curious if these embeddings are equally 'well behaved' as we move up the dimensional hierarchy or if low dimensional embeddings are somehow 'better behaved' than higher dimensional ones. Any pointers are appreciated.
 A: If you replace $[\mathcal{C},\mathfrak{cat}]^\uparrow$ by the 2-category of strict functors $\mathcal{C} \to \mathfrak{cat}$, strict natural transformations, and modifications (whose underlying 1-category is your $[\mathcal{C},\mathfrak{cat}]^\uparrow$), then the resulting 2-functor (I'm not sure what you mean by "2-embedding") has both a left and a right adjoint.
Abstractly, this is an instance of the functor $T\text{-Alg}_s \to T\text{-PsAlg}$ from the 2-category of strict algebras (and strict morphisms) for a strict 2-monad to its category of pseudoalgebras (and pseudomorphisms), which has a left adjoint (the "pseudomorphism classifier") whenever $T$ is an accessible 2-monad on a locally presentable 2-category.  It's also an instance of the dual situation for a 2-comonad; hence we get both left and right adjoints.
For higher values of $n$, in general there won't even be an $n$-category of $n$-categories.  For instance, bicategories, pseudofunctors, and pseudonatural transformations don't form a bicategory; they only satisfy the interchange law up to a 3-cell.  (There is a 2-category of bicategories, pseudofunctors, and icons, but that may not be what you have in mind.)
