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Let $\mathcal C$ be a strict, locally small 2-category. Consider a subcategory $\mathcal L$ of $\mathcal C$ such that $\mathcal L$ has the same objects as $\mathcal C$, and the arrows of $\mathcal L$ are the left adjoints in the 2-category $\mathcal C$. Let $U:\mathcal L\to\mathcal C$ be the inclusion functor.

When is $U$ a (left or right) adjoint?

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    $\begingroup$ If $\mathcal{L}$ is discrete and $\mathcal{C}$ is not, then $U$ cannot have any adjoint. For instance, this happens when $\mathcal{C}$ is a poset considered as a (locally discrete) 2-category. But sometimes $U$ has a right adjoint: for example, when $\mathcal{C} = \mathbf{Rel}$ (considered as a locally posetal 2-category) and $\mathcal{L} = \mathbf{Set}$. $\endgroup$
    – Zhen Lin
    Mar 5, 2014 at 13:25
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    $\begingroup$ @ZhenLin, yes. ${\mathbf{Set}}\subseteq{\mathbf{Rel}}$ is my motivating example. It is clear to me that, in some sense, $\mathcal C$ must have "enough" left adjoints but I am getting lost in the arrow swamp that emerges from my pen whenever I try to express this idea. Maybe I need a better pen. Or a better brain. $\endgroup$ Mar 5, 2014 at 13:42
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    $\begingroup$ I don't think that's enough. The right adjoint for $\mathbf{Set} \hookrightarrow \mathbf{Rel}$ is the powerset functor, which means closely related examples like $\mathbf{Set}_{< \aleph_1} \hookrightarrow \mathbf{Rel}_{< \aleph_1}$ will fail to have a right adjoint. $\endgroup$
    – Zhen Lin
    Mar 5, 2014 at 13:53
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    $\begingroup$ Some background you might already know: An inclusion of 2-categories $B \to C$ such that every 1-cell of $B$ has a right adjoint in $C$ is called a proarrow equipment. Examples are the inclusions $\mathbf{Set} \to \mathbf{Rel}$ and $\mathbf{Cat} \to \mathbf{Prof}$ ($\mathbf{Prof}$ = profunctors); in the latter case $\mathbf{Cat}$ contains all left adjoint profunctors if we restrict to Cauchy-complete categories. A structure yielding powersets in the former case and presheaves in the latter case is a Yoneda structure. Size issues prevent presheaves from forming a legitimate left adjoint. $\endgroup$
    – Tim Campion
    Mar 5, 2014 at 17:02
  • $\begingroup$ @TimCampion Thanks for the keywords, very helpful. $\endgroup$ Mar 5, 2014 at 19:24

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Your question is impossible to answer in full generality. There are, however, some interesting special cases. I shall focus on your motivating example (I think, I have written about this on many occasions, but cannot recall the exact sources at the moment).

If $\mathbb{C}$ is a (finitely complete, locally small) regular category, then you may construct a category $\mathit{Rel}(\mathbb{C})$ of canonical internal relations in $\mathbb{C}$. Moreover, $\mathit{Rel}(\mathbb{C})$ has a 2-categorical structure induced by the natural ordering of monomorphisms, and $\mathbb{C}$ is equivalent to the (2-)subcategory of $\mathit{Rel}(\mathbb{C})$ consisting of morphisms that have right adjoint.

On the other hand, the inclusion: $$J \colon \mathbb{C} \rightarrow \mathit{Rel}(\mathbb{C})$$ has a right adjoint: $$P \colon \mathit{Rel}(\mathbb{C}) \rightarrow \mathbb{C}$$ iff $\mathbb{C}$ is an elementary topos. Indeed, one may easily verify, that $P$ has to be the internal power functor: $$\hom_{\mathit{Rel}(\mathbb{C})}(A, B) \approx \hom_\mathbb{C}(A, P(B))$$

More generally, if $p$ is a regular fibration, then under some mild conditions, $p$ has a generic object iff the canonical embedding: $$J \colon \mathit{Map}(\mathit{Rel}(p)) \rightarrow \mathit{Rel}(p)$$ has a right adjoint, where $\mathit{Rel}(p)$ is the category of $p$-internal relations, and $\mathit{Map}(\mathit{Rel}(p))$ is the subcategory of $\mathit{Rel}(p)$ consitiong of morphisms that have right adjoints.

The link with the previous example appears, when we consider for $p$ the canonical subobject fibration $\mathit{sub}_\mathbb{C}$ of a regular category $\mathbb{C}$ (the canonical subobject fibration of a regular category is regular, and the generic object of a subobject fibration corresponds to the subobject classifier of the category). Another interesting case, is when instead of a subobject fibration, we consider a fibration of regular subobjects --- which leads to the definition of a quasitopos.

Another view of the above generalization is to consider an allegory $\mathbb{A}$. The subcategory $\mathit{Map}(\mathbb{A})$ of $\mathbb{A}$ consisting of maps that have right adjoints is called the category of maps in $\mathbb{A}$. An allegory $\mathbb{A}$ is a "power allegory" iff the canonical inclusion: $$J \colon \mathit{Map}(\mathbb{A}) \rightarrow \mathbb{A}$$ has right adjoint $P$.

As you may see, your question for mere 2-posetal categories (i.e. $\hom(X, Y)$ is a poset for every $X, Y$) of relations is highly non-trivial.

In some sense, the concept of proarrow equipment (pointed by Tim in his comment) is a 2-categorification of the concept of an allegory. Category $\mathbf{Cat}$ can be reconstructed, up to Cauchy completion (one should not expect more by categorical methods, because $\mathbf{Prof}$ does not distinguish between a category and its Cauchy completion --- i.e. they are equivalent in $\mathbf{Prof}$), from $\mathbf{Prof}$ by taking profunctors that have right adjoints. The inclusion: $$J \colon \mathbf{Cat}_\mathit{CC} \rightarrow \mathbf{Prof}$$ does not have right adjoint due to the size issues. The same is true for the inclusion: $$J \colon \mathbf{Cat}_\mathit{CC}(\mathbb{C}) \rightarrow \mathbf{Prof}(\mathbb{C})$$ of Cauchy complete $\mathbb{C}$-internal categories to $\mathbb{C}$-internal profunctors, for any non-trivial category $\mathbb{C}$.

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