I've been learning about sheaves and have, I think, understood its characterisation
as a space that glues together locally, and its presentation as a space over a base to which it is locally homeomorphic.


However I've come across the following description of this equivalence (Tom Leinster, Sheaves do not belong to algebraic geometry, <http://www.maths.gla.ac.uk/~tl/sheaves.pdf>) 
which is induced from an adjunction using general categorical ideas. His description doesn't mention either local homeomorphisms or the sheaf condition. Obviously they must be there, but I don't see how to extract it.


In brief: He uses the adjunction 


$Cat[A^{op},Set]$<->$B$


where $A$ is a small category, $B$ has small colimits, and we have
a functor $I:A \to B$, then 


$\leftarrow$ is $B[I,-]$ and
$\rightarrow$ is -(tensor)I which IS defined as the left adjoint, which he states
 exists as a particular colimit


He then looks at the restriction of the adjunction where it becomes an equivalence, then setting for an X in Top, $A:=Opn(X)$, $B:=Top/X$
and $I:A\to B=Opn(X) \to Top/X$ the obvious functor, then the adjunction


$PShf X=Cat[Opn(X)^{op},Set]$<->$Top/$X restricts to equivalence $Shf X$ <-> $Et X$


I Just wanted to note that a similar question has been posted on Math Overflow before:  

[Understanding the etale space construction from a formal viewpoint](https://mathoverflow.net/questions/36016/)


and one of the answers suggests looking at Sketches of an Elephant: A Topos Theory Compendium by Johnstone, which is what I'm going to do



Extracting the Sheaf and espace étalé condition from an abstractly given equivalence between these two spaces
The general definition of sheaf, is much more general than a local homeomorphism over a space- however, it is sheaves of this later kind that came about first. In the general framework, you have a small category $C$ and a $\textbf{Grothendieck topology}$ on $C$, which declares what families of arrows $\left(c\_i \to c\right)$ are $\textbf{covering families}$. The sheaf condition for open covers has an obvious generalization in this setting. Sheaves form a reflective subcategory of presheaves, and the reflector, is given by the "sheafification" functor. In the classical setting of sheaves over a space, $C$ is the poset of open subsets of a space $X$, and the Grothendieck topology is given by open covers. There is a canonical functor $Open\left(X\right) \to Top/X$, and by general abstract nonsense, this produces an adjunction between $Psh\left(Open\left(X\right)\right)$ and $Top/X$. The essential image of $$Psh\left(Open\left(X\right)\right) \to Top/X$$ is those maps $Y \to X$ which are local homeomorphisms, and this functor is the "espace étalé" construction. The other functor, sends a map $Y \to X$ to its sheaf of sections. Starting with a presheaf $F$, constructing its espace étalé, and then taking it sections, yields a sheaf. This composition if a functor $Psh\left(Open\left(X\right)\right) \to Sh\left(X\right)$, and is in fact the sheafification functor. You can read more about this in, for instance, "Sheaves in Geometry and Logic".


