I am studying the same book (Perhaps different edition). Here is what I think author intended. Consider category Sets. Suppose $f: A \rightarrow B$. Now consider point $1 \rightarrow A$. A varying element V has following properties:
$$V(1 \rightarrow A)=1 \rightarrow B$$
(It appears that $V$ is a map from set of points of $A$ to set of points of $B$.)
This can be shown by composing points with $f$ as follows:
Where $a: 1 \rightarrow A$ and $b:1 \rightarrow B$.
So evaluating $V$ at different points of $A$ gives a family of points of B. For every point of $A$, $f$ assigns a point of $B$. This is one perspective for a map. Now, we can replace 1 by D, and say that a map $f:A \rightarrow B$ is a family of D-elements of B.
What I do not understand though, is last sentence of that paragraph, which is:
For example, we can take D = A, and the
identity $D \rightarrow A$, thus revealing that (c) the varying element, as a single thing, is a
single figure or element itself.
Any ideas on this?