Mackey coset decomposition formula

Question

I have a question about following argument I found in these notes on Mackey functors:

(2.1) LEMMA. (page 6) Let $G$ be a finite group and $J$ any subgroup. Whenever $H$ and $K$ are subgroups of $J$, there is a pullback diagram of $G$-sets

$$ \require{AMScd} \begin{CD} \Omega @>{} >> G/K \\ @VVV @VVV \\ G/H @>{}>> G/J \end{CD} $$

where we have Mackey double coset decomposition $\Omega= \coprod_{x \in [H\backslash J/K]}G/(H \cap x^{-1}Kx)$

The proof shows by hand for the special case $G=J$ that the pullback in

$$ \require{AMScd} \begin{CD} J/K \times J/H @>{} >> J/K \\ @VVV @VVV \\ J/H @>{}>> J/J = \{*\} \end{CD} $$

is given by decomposition $J/K \times J/H= \coprod_{x \in [H\backslash J/K]}J/(H \cap x^{-1}Kx)$ and then claims then claims that the result follows from application of the usual induction functor $\text{Ind}^G _J: (J-\text{Set}) \to (G-\text{Set})$ sending a $J$-set $S$ to $G$-set $G \times_J S$.

Question: Why this functor preserves limits (eg a pullback as in this case). It is well known that the induction functor $\text{Ind}^G _J$ is left-adjoint to usual restriction functor $\text{Res}^J _G$, and therefore definitely preserves colimits. But the pullback in the question is a limit, so I not understand why the induction functor applied to $J/K \times J/H$ from the special case preserves the limit/ pullback property.

A remark: I found googling this problem several alternative proofs of the claimed decomposition formula, but the objection of this question is really to understand why this argument given by Webb in the linked notes works, ie why in this situation the induction functor commutes with the pullback.

Answer 1

The induction functor preserves all (weakly) connected limits (note that this covers pullbacks, but not products - and of course, because it doesn't preserve the terminal object, it couldn't do both !).

There is a more general way to argue, but in this case you can say as follows (there is also probably a more concrete one) : one can check on underlying objects whether this preserves a given limit diagram. But forgetting the $G$-structure on $Ind_J^G$ is equivalent to forgetting the $J$-structure and taking a $G/J$-indexed constant coproduct. Thus it suffices to check that coproducts on the category of sets commute with weakly connected limits.

To prove this, note that for a given set $I$ (the indexing set), the equivalence $Set^I \simeq Set_{/I}$ intertwines the "coproduct" functor and the forgetful functor.

But now it is a general fact that if $C$ has $K$-shaped limits, and $K$ is weakly connected, then the forgetful functor $C_{/x} \to C$ preserves $K$-shaped limits.