TITLE: Kan extensions in the $2$-category of monoidal categories
QUESTION [12 upvotes]: Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of $k$-linear categories, $k$-linear functors, natural transformations. Where have Kan extensions been studied and used explicitly for other $2$-categories (a related question was asked here, but this doesn't really go into examples)? In particular I am interested in the $2$-category of monoidal categories, lax monoidal functors and lax monoidal natural transformations, also with "strong" instead of "lax". For example, the left Kan extension of a lax monoidal functor $(f,\eta,\mu) : I \to A$ (between monoidal categories $I,A$) along the unique lax monoidal functor $I \to \{1\}$ is a universal monoid object $M=(X,\eta,\mu)$ in $A$ equipped with a cocone $\{\alpha_i : f(i) \to X\}_{i \in \mathrm{Ob}(I)}$ which is lax monoidal in the sense that (a) $1 \xrightarrow{\eta} f(1) \xrightarrow{\alpha_1} X$ equals $1 \xrightarrow{\eta} X$ and (b) the diagram
$$
\begin{array}{ccc}
f(i \otimes j) & \xrightarrow{\alpha_{i \otimes j}} & X \\ {\scriptsize\mu}\uparrow ~~&&~~\uparrow{\scriptsize\mu} \\ f(i) \otimes f(j) & \xrightarrow{\alpha_i \otimes \alpha_j} & X \otimes X
\end{array}
$$
commutes. Is there any interesting example for this kind of "lax monoidal colimit"? If yes, what does it mean for a monoidal category to be "lax monoidal cocomplete"? Consider for example the case that the underlying category of a monoidal category is cocomplete and the tensor product is cocontinuous in each variable (this is what I would call a cocomplete monoidal category, or perhaps more precisely, a monoidal cocomplete category, because this is precisely a monoid in the symmetric monoidal $2$-category of cocomplete categories), which is quite typical. Do "lax monoidal colimits" exist then? What about "lax monoidal left Kan extensions" in general or "strong monoidal left Kan extensions" in general?

REPLY [3 votes]: I am not an expert, but I think the best reference for this topic is Algebraic Kan extensions along morphisms of internal algebra classifiers by Mark Weber.