The direct image sheaf vs. the pushforward of the Etale space This is something that came up when I was trying to prove the equivalence of categories of Etale spaces and sheafs over $X$ to myself.
Let's start with a sheaf $F$ over $X$ with an associated étalé space $E$ with projection $\pi: E \to X$. Now imagine we have a map $f: X \to Y$. Then there are two seemingly natural ways to construct a direct image sheaf.
The first is the usual definition of:
$$ f_*F(U) = F(f^{-1} (U)) $$
but you can also use the map $f \circ \pi: E \to Y$ and look at the sheaf of (local) sections $\Gamma(E/Y)$.
This latter construction has the useful property that there's a natural map from sections of $f \circ \pi$ to sections of $f$ by composing with $\pi$, so you get a morphism of sheafs $\Gamma(E/Y) \to \Gamma(X/Y)$ (which is especially interesting in the special cases where $X$ is itself an étalé space over $Y$)
However I can't get the two to agree. Since $E$ is an étalé space we have that $f_*F(U) = \Gamma(E/X)(f^{-1}(U))$ which consists of sections $s: f^{-1}(U) \to E$ of $\pi$, but I can't manage to figure out how to map this onto a corresponding section $t : U \to E$ in $\Gamma(E/Y)(U)$. I seem to be missing a natural right inverse of $f$.
So my question is are $f_*F$ and $\Gamma(E/Y)$ the same, and why (not)?
 A: For a topological space $X$, let $\textbf{Sh} (X)$ be the category of sheaves of sets on $X$.
Let $\textbf{LH}$ be the category of topological spaces and local homeomorphisms.
You know that the "sheaf of sections" functor $\Gamma : \textbf{LH}_{/ X} \to \textbf{Sh} (X)$ is fully faithful and essentially surjective on objects.
In fact, it is (pseudo)natural in the sense that, given any continuous map $f : X \to Y$, $f^* \Gamma$ and $\Gamma f^*$ are isomorphic as functors $\textbf{LH}_{/ Y} \to \textbf{Sh} (X)$, in a way that is coherent with composition.
There is a "direct image" functor $f_* : \textbf{Sh} (X) \to \textbf{Sh} (Y)$ right adjoint to the "inverse image" functor $f^* : \textbf{Sh} (Y) \to \textbf{Sh} (X)$, defined by the formula you mention.
It is not so easy to describe as a functor $\textbf{LH}_{/ X} \to \textbf{LH}_{/ Y}$, but for formal reasons it exists.
On the other hand, when $f : X \to Y$ is itself a local homeomorphism, the pullback functor $f^* : \textbf{LH}_{/ Y} \to \textbf{LH}_{/ X}$ has a left adjoint, denoted variously as $f_!$ or $\Sigma_f$, defined simply as sending a local homeomorphism $\pi : E \to X$ to $f \circ \pi : E \to Y$.
It is not as easy to describe as a functor $\textbf{Sh} (X) \to \textbf{Sh} (Y)$, but anyway it is (in general) different from $f_* : \textbf{Sh} (X) \to \textbf{Sh} (Y)$.
The easiest way to get a feel for the difference between $f_!$ and $f_*$ is to look at the case where $X$ and $Y$ are discrete sets.
Then a sheaf $F$ on $X$ is just an $X$-indexed family of sets, say $( F_x : x \in X )$, and:
$$(f_! F)_y = \coprod_{x \in f^{-1} \{ y \}} F_x$$
$$(f_* F)_y = \prod_{x \in f^{-1} \{ y \}} F_x$$
In particular, a key difference is that if $y$ is not in the image of $f : X \to Y$, then $(f_! F)_y$ is empty but $(f_* F)_y$ is a singleton.
