fiber product of schemes

Hi

suppose one have the following situation
A <--- AxB ---> B
| |
v v
X <--- XxY ---> Y
with schemes A,B,X and Y. The symbol x denotes the fiber product and
the horizontal maps are the projections respectively. One gets a map
f:AxB --> XxY by the universal property of the fiber product. Suppose
the vertical maps have a property which is closed under base change
and composition ('closed immersion' or 'proper' for example). Does the
induced map f necessarily have this property too?

Yours
Alex
.

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