ct.category theory - If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits?

Motivation

Suppose that $Fcolon Xto A$ is left adjoint to $Gcolon Ato X$, and let
$varepsiloncolon FGstackrel{.}{to}I_A$ be the counit of the adjunction.
Suppose also that $A$ is $J$-complete (for some category $J$), so that
$operatorname{Lim}$ is a functor $C^Jto C$, where for an arrow
$alphacolon T_1stackrel{.}{to} T_2$ 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 $Tcolon Jto A$ be a functor. We have the natural transformation
$varepsilon Tcolon FGTstackrel{.}{to} T$, 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
$taucolon operatorname{Lim}(T)stackrel{.}{to}T$ is the lower limiting cone,
then $FGtaucolon FGoperatorname{Lim}(T)stackrel{.}{to}FGT$ is the upper
limiting cone, and the above diagram becomes

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

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

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 $Fcolon Xto A$ is left adjoint to $Gcolon Ato X$ 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.