categories: Subclassifier object question

[Galchin Vasili <vngalchin@yahoo.com>]

Hello,

    First of all sorry for the beginner's question. I
have been reading various sources about the notion
of a subobject classifier, e.g. McClarty's book, 
Lawvevre's book. Let assume the following situation: 


1) Has a subobject classifier, 
       true : 1 ---------> Omega.

2) Have an object X and object A and
     a monomorphism subX: A------->X.

3) With 2), we will have the pullback diagram
     with the corners of the pullback
     being the subobject classifier
     and the unqiue classifier/classifier
X------>Omega

4)We have either a element, el: 1------>X or a 
     generalized el: element H------->X.


Question: Let's assume for simplicity that our
category
   is Set. The problem is I don't understand how the
   notion of a subobject classifier determines whether
   "el" does or does not belong to the aforementioned
   subobject, "subX"! (I know that the outer commuting
   square figures in this, but it seems like every
   element will be "classified" as belonging 
   to "subX").


Thanks and regards, Bill Halchin