*All sets and groups in the question are finite.*

In order to understand equivariant sheaves better **I'm trying to prove some basic facts from Mackey theory using equivariant sheaves.** The main obstacle i've been faced with is the difficulty of keeping track of all the different equivalences. 

Let $G$ be a group and $K \lt G$ a subgroup. Perhaps one of the most basic canonical equivalences is between $Sh_G(G/K)$ and $Sh_K(pt)$. This can be implemented as follows:

> Let $p: G \to pt$ and $q : G \to G/K$ be the projections. Then one has
> the following compositions:
> 
> $$Sh_K(pt) \overset{k \mapsto k^{-1}}{\longrightarrow} 
 Sh_{K^{op}}(pt) \overset{p^*}{\to} Sh_{G \times K^{op}}(G)
 \overset{q^{K^{op}}_*}{\to} Sh_G(G/K)$$
> 
> Explanations:
> 
>  1. First arrow is the inversion isomorphism from $K$ to $K^{op}$. It is obviously an equivalence.
>  2. The second arrow is the pullback which remembers the left $G$-equivariant structure coming from the fact that the projection
> $G/K \to pt$ is left $G$-equivariant. This is an equivalence since the
> $G$-action is free.
>  3. The third arrow is $K^{op}$-invariant sections of the pushforward along $q: G \to G/K$. This is an equivalence since the right $K^{op}$ action
> is free.
> 
> The functor in the other direction is the composition of the pseudo
> inverses.

This is all very neat until one tries to use this equivalence to prove stuff. I'll try to explain with an example. **Suppose we wanted Mackey's induction formula.** In other words **we'd like to describe the functor** $Res^K_G \circ Ind^G_H$ in terms of other functors. 


Let $p: G/H \to pt$ and $q: G/K \to pt$. In this notation the composition above admits the following description:

$$Sh_G(G/H) \overset{p_*}{\to} Sh_G(pt) \overset{Res^K_G}{\to} Sh_K(pt)$$

By commutativity of the corresponding diagram this composition is isomorphic to the composition:

$$Sh_G(G/H) \overset{Res^K_G}{\to} Sh_K(G/H) \overset{p_*}{\to}  Sh_K(pt)$$

Once one realizes that the category $Sh_K(G/H)$ splits as a direct sum  $\bigoplus_{\mathcal{O}} Sh_K(\mathcal{O})$ over all $K$-orbits in $G/H$, then the **second functor above** $p_*$ becomes easy to describe as it is just a **direct sum of induction functors.**

> **However**, the following functor:
> 
> $$F: Sh_K(pt) \to Sh_G(G/H) \overset{Res^K_G}{\to} Sh_K(G/H) \to
 \bigoplus_{\mathcal{O}} Sh_K(\mathcal{O})$$ 
> 
> Obtained by composing the first functor with equivalences **seems rather
> mysterious.**

It looks like it should be a **sum over certain restriction functors** (a fact we already know if we know about the classical mackey formula). Beyond that however **this functor is completely mysterious to me**. In order to compute it I found no other choice other than c**omposing all the functors** involved and which resulted in **a huge expression I didn't manage to decipher.** 


> **How can one describe $F$ using the language of equivariant sheaves?** 
>
>  **How does one compute similar functors in practice?**

Of course I already know what $F$ should be from the classical Mackey formula but i'm looking for a way to "grab hold" of this functor using the language equivariant sheaves.