Visualizing a homotopy pull back I am currently taking a course in algebraic topology, which also covers a lot of category theory. My question is pretty straightforward:

How do you visualize the (homotopy) pull back of a diagram $B\to
 C\leftarrow A$ ?

In class the professor usually just kind of says "Well if you think about it you get....".  I have figured out that for a (homotopy) push out you want to glue the two spaces together along their common points but the description of the pullback has eluded me.
 A: The standard construction of the homotopy pullback involves the total path space $C^I$ (i.e. the space of all continuous maps $I \to C$, with the appropriate topology). More precisely, if $f : A \to C$ and $g : B \to C$ are the given maps, then:
$$A \times^\mathrm{h}_C B = \{ (a, p, b) \in A \times C^I \times B : p(0) = f(a), p(1) = g(b) \}$$
The path space $C^I$ is rather difficult to visualise, as it is generally infinite dimensional. But the idea should be clear enough: a point of $A \times^\mathrm{h}_C B$ is a point in $A$ and a point in $B$ together with a path in $C$ connecting the images of those two points.
Notice that there is a canonical map $A \times_C B \to A \times^\mathrm{h}_C B$. Under good conditions, this is a homotopy equivalence. Indeed, if $g : B \to C$ is a Hurewicz fibration, then the homotopy extension/lifting property gives us a map $(A \times^\mathrm{h}_C B) \times I \to B$ extending the canonical map $(A \times^\mathrm{h}_C B) \times \{ 1 \} \to B$ and lifting the canonical map $(A \times^\mathrm{h}_C B) \times I \to C$; restricting to $(A \times^\mathrm{h}_C B) \times \{ 0 \} \to B$ we then get a commutative square
$$\begin{array}{ccc}
(A \times^\mathrm{h}_C B)  & \rightarrow & B \\
\downarrow &  & \downarrow \\
A & \rightarrow & C 
\end{array}$$
and hence a map $A \times^\mathrm{h}_C B \to A \times_C B$ that is left inverse to the canonical map $A \times_C B \to A \times^\mathrm{h}_C B$; in fact, this map $A \times^\mathrm{h}_C B \to A \times_C B$ is also a homotopy right inverse. Thus, when $g : B \to C$ is a Hurewicz fibration, the homotopy pullback coincides with the ordinary pullback (up to homotopy equivalence).
