Thanks Owen, you explained it pointedly.

Intuitively, the [category of elements](, that specializes the Grothendiek construction (as if he hasn't construed anything more!) would respond to the question "how do I turn arrows in a plain old Venn diagram picture of a function between sets into **morphisms** in a good category? (here I write two functions to spice it up).

![Venn diagram](

Since we have a concrete category \\(I:\mathcal{C} \to Set\\), the objects can be viewed ("correspond") to actual sets, and the morphisms to functions. So if \\(\mathcal{C}\\) has objects say \\(S, T, U\\), and morphisms \\(f, g\\), then \\(Gr(I)\\) will have, for its objects, the disjoint union of all the sets that correspond to the objects of \\(\mathcal{C}\\). Hence the objects of \\(Gr(I)\\) form a soup of all the _elements_ of the objects of \\(\mathcal{C}\\), with their object filiation flattened and forgotten, but then _regained_ by means of \\(\pi:Gr(I) \to \mathcal{C}\\).

Here is what \\(\pi\\) does to the objects and morphisms of \\(Gr(I)\\), painted with dashed arrows.


The category of elements is an incomplete part of the story, the end product is the functor \\(\pi\\), that says how \\(Gr(I)\\) sits on top of \\(\mathcal{C}\\). \\(\pi\\) organizes the data.