Thanks Owen, you explained it pointedly.

Intuitively, the [category of elements](https://ncatlab.org/nlab/show/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](http://i66.tinypic.com/v7xt2t.jpg)

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.

![gro-const](http://i64.tinypic.com/30nhzly.jpg)

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.