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Reading this title, you may have thought there was a typo, but there isn't (well I don't think there is at least!). This question arises from a definition I formulated recently, and would like to understand better. Perhaps a more seasoned category theorist could take a stab at it, or point me in the right direction.

$\Large{Background}$

${\large Definition:}$ A category consists of a class of morphisms and a class of objects, which satisfy the usual properties. (I just want to stress the class part of it.)

${\large Definition:}$ Consider the metacategory\footnote{this follows Mac Lane's definitions, or is just a category in the modern parlance.} $\underline{Cat}$ of categories. Consider a (meta)functor $\mathfrak{i} : \underline{Cat} \rightarrow \underline{Cat}$ which preserves adjoints, i.e. if $F \dashv G$, then $\mathfrak{i} F \dashv \mathfrak{i} G$; consider a natural transformation $P_{-}^{\mathfrak{i}} : id_{\underline{Cat}} \rightarrow \mathfrak{i}$. For any category $\mathfrak{i}$, we define a left adjoint to ${P_{\mathfrak{C}}^{\mathfrak{i}}} : \mathfrak{C} \rightarrow \mathfrak{i} \mathfrak{C}$ to be a universal universal construction, and a right adjoint to $P^{\mathfrak{i}}_{\mathfrak{C}}$ to be a universal couniversal construction. We typically will abbreviate these as UUCs and UCCs.

Why this definition? I formulated this definition, because it is exactly the property that makes the following proposition work. Also, you can talk about a given UUC or UCC in any category $\mathfrak{C}$, like you can with the limit of a diagram in an arbitrary category, regardless of whether it exists or not. Thus a universal construction that arises in this way seems more "universal" to me, at least. I can discuss this in greater detail if anybody wants me to, but it isn't all that essential to the problem.

${\large Proposition:}$ Consider $F:\mathfrak{C}\rightarrow\mathfrak{D} \in \underline{Cat}$. If $L\dashv F$ and a given UCC $(\mathfrak{i}, P_{-}^{\mathfrak{i}})$ exists in both $\mathfrak{C}$ and $\mathfrak{D}$, then the UCC commutes in $\mathfrak{D}$. That is, if $P_{\mathfrak{C}}^{\mathfrak{i}} \dashv {\mathcal{R}}_{\mathfrak{C}}^{\mathfrak{i}}$ and $P_{\mathfrak{D}}^{\mathfrak{i}} \dashv {\mathcal{R}}_{\mathfrak{D}}^{\mathfrak{i}}$, then $\mathcal{R}_{\mathfrak{D}}^{\mathfrak{i}}\circ\mathfrak{i} F \cong F \circ \mathcal{R}_{\mathfrak{C}}^{\mathfrak{i}}$. A similar result holds for all manner of permutations of left and right adjoints.

(I have an elaborate xymatrix diagram depicting the simple proof, but I don't think it works with jsmath...)

${\Large Question}$

Are the only UUCs and UCCs just colimits and limits respectively, if we take the initial and terminal object constructions as a special case of a (co-)limit?

${\large Remark:}$ I wasn't able to come up with any other examples, and the various adjoint functor theorems suggest that limits and colimits are the essential constructions that are needed for there to be an adjoint, which led me to believe the answer to my question is that limits and colimits are the only UCCs and UUCs. I must admit that I don't have a great understanding of the proofs of Freyd's Adjoint Functor Theorem or the SAFT, so perhaps the essential tool is stuck in there and I am just being too lazy. =)

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I'm in the process of thinking about how I should formalize when two UUCs or UCCs are the same. I'm a bit busy though, so it might take me a while. –  Glen M Wilson Jun 9 '10 at 11:18
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