I have a question about following argument I found
in [these notes](https://www.google.de/url?sa=t&source=web&rct=j&opi=89978449&url=https://people.math.rochester.edu/faculty/doug/otherpapers/WebbMF.pdf&ved=2ahUKEwiiu9Lu7d6AAxVg_rsIHU-KCSEQFnoECBAQAQ&usg=AOvVaw1OZ96p6a5V9Y-wW04DRo45) 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.



Mackey coset decomposition formula
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.


