## Motivation

Suppose that $F\colon X\to A$ is left adjoint to $G\colon A\to X$, and let $\varepsilon\colon FG\stackrel{.}{\to}I_A$ be the counit of the adjunction. Suppose also that $A$ is $J$-complete (for some category $J$), so that $\operatorname{Lim}$ is a functor $C^J\to C$, where for an arrow $\alpha\colon T_1\stackrel{.}{\to} T_2$ of $C^J$, $\operatorname{Lim}(\alpha)$ is the unique arrow of $A$ for which the following diagram is commutative:

$$ \begin{matrix} \operatorname{Lim}(T_1)& \stackrel{\text{limiting cone}}{\longrightarrow} & T_1\\ | & & |\\ \operatorname{Lim}(\alpha) & & \alpha\\ \downarrow & & \downarrow \\ \operatorname{Lim}(T_2)& \stackrel{\text{limiting cone}}{\longrightarrow} & T_2 \end{matrix} $$

Let $T\colon J\to A$ be a functor. We have the natural transformation $\varepsilon T\colon FGT\stackrel{.}{\to} T$, and $\operatorname{Lim}(\varepsilon T)$ is the dotted line making the following diagram commutative:

$$ \begin{matrix} \operatorname{Lim}(FGT)& \stackrel{\text{limiting cone}}{\longrightarrow} & FGT\\ | & & |\\ \operatorname{Lim}(\varepsilon T) & & \varepsilon T\\ \downarrow & & \downarrow \\ \operatorname{Lim}(T)& \stackrel{\text{limiting cone}}{\longrightarrow} & T \end{matrix} $$

If $FG$ preserves $J$-limits, and $\tau\colon \operatorname{Lim}(T)\stackrel{.}{\to}T$ is the lower limiting cone, then $FG\tau\colon FG\operatorname{Lim}(T)\stackrel{.}{\to}FGT$ is the upper limiting cone, and the above diagram becomes

$$ \begin{matrix} FG\operatorname{Lim}(T)& \stackrel{FG\tau}{\longrightarrow} & FGT\\ | & & |\\ \operatorname{Lim}(\varepsilon T) & & \varepsilon T\\ \downarrow & & \downarrow \\ \operatorname{Lim}(T)& \stackrel{\tau}{\longrightarrow} & T \end{matrix} $$

Since the naturality of $\varepsilon$ implies that for all $j\in \operatorname{obj}(J)$ the diagram $$ \begin{matrix} FG\operatorname{Lim}(T)& \stackrel{FG\tau_j}{\longrightarrow} & FGT(j)\\ | & & |\\ \varepsilon_{\mathrm{Lim}T}& & \varepsilon_{T(j)}\\ \downarrow & & \downarrow \\ \operatorname{Lim}(T)& \stackrel{\tau_j}{\longrightarrow} & T(j) \end{matrix} $$

is commutative, it follows that $\varepsilon_{\mathrm{Lim}T}$
can replace $\operatorname{Lim}(\varepsilon T)$ in the last but one
diagram while keeping it commutative. By uniqueness, we get
the nice equation

$$
\varepsilon_{\mathrm{Lim}T} = \operatorname{Lim}(\varepsilon T).
$$
Note that it seems that all depends on $FG$ preserving $J$ limits.

## Question

If $F\colon X\to A$ is left adjoint to $G\colon A\to X$ and $A$ has $J$-limits, when does $FG$ preserve $J$-limits? This is obviously true when $F$ preserves limits (for example, when there is also a left adjoint to $F$), but are there other interesting situations?

## Background

For solving an exercise from Mac Lane, I used some
results from A. Gleason, ''Universally locally connected
refinements,'' Illinois J. Math, vol. 7 (1963), pp. 521--531. In that
paper, Gleason constructs a right adjoint to the inclusion functor
$\mathbf{L\ conn}\subset \mathbf{Top}$ ($\mathbf{L\ conn}=$ locally
connected spaces with continuous maps), and proves that the counit

of the product of two topological spaces is the product of the
counits (Theorem C). This made me curious when do counits
and limits interchange.

leftadjoint. (Think of CpctHff into RegularTop, or AbGp into Gp ...) So something atypical seems to be going on, unless I've misunderstood (which might admittedly very well be the case) $\endgroup$coreflective subcategory (the inclusion functor has arightadjoint). Searching the web, I found some additional results on coreflective subcategories in general topology in H. Herrlich and G.E. Strecker, "Coreflective subcategories in general topology," Fund. Math. 73 (1972), pp. 199--218 (matwbn.icm.edu.pl/ksiazki/fm/fm73/fm73124.pdf). Mac Lane also gives an example from algebra: torsion abelian groups in abelian groups. $\endgroup$