ined as a composite map
|
| U_1  -&gt;  U_1 x U_2  -&gt;  V_1 x V_2  -&gt;  V_1,
|
| the first map being an inclusion and the third a projection.
|
| We have therefore defined a C^p-structure on Y which will be called
| a &quot;submanifold&quot; of X.  This structure satisfies a universal mapping
| property, which characterizes it, namely:
|
| | Given any map f : Z -&gt; X from a manifold Z into X such that
| | f(Z) is contained in Y.  Let f_Y : Z -&gt; Y be the induced map.
| | Then f is a morphism if and 