co-exponential question

["Bill Halchin" <bhalchin@hotmail.com>]

This is actually a "dual" question.
Basically I want to do the dual of the construction
gives the notion of an exponential or map object.

Suppose we have a category C with sums. Then we build the
following category from C.

object:  T+X<-----Y

map: from T+X<-----Y  to T+X'<-------Y is a C map "alpha"  such
that we have the following diagram:

                   I-sub-T+alpha
         T+X---------------------------->T+X'
           ^                              ^
            \                            /
             \                          /
              \                        /
               \                      /
                \                    /
                 \                  /
                  \                /
                   \              /
                    \            /
                     \          /
                      \        /
                       \      /
                        \    /
                         \  /
                          \/
                          Y


   Then suppose there exists a C-object called Y**T such
that  T+Y**T<-------Y is the initial object of the category
just built above. What significance does Y**T have opposed
to the concept of an exponential???? If I did everything
correctly it (Y**T) should be the dual of T**Y.


Regards,

Bill Halchin