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
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