I tried 3 different examples and instead of just choosing 2 subsets as the exercise asked for, I solved it for every subset since 2 and 3 element power sets aren't too big to draw out.
![2-3 Pullback ver1](http://aether.co.kr/wp-content/uploads/2-3-pullback-f1.jpg)
![2-3 Pullback ver2](http://aether.co.kr/wp-content/uploads/2-3-pullback-f2.jpg)
I've been struggling with this all week and still can't seem to figure this one out. I think I am having a problem getting my head around the logical expressions in the definitions of \\(f_!\\) and \\(f_*\\). So instead I just used the definitions of left and right adjoints got the answers above. Not sure if this is correct but they preserve order and seem to obey all the definitions of an adjoint. But what doesn't make sense for me is the intuitive explanations of these functions given in by Spivak and Fong.
>[Left Adjoint] hence takes a set A of balls, and tells you all the buckets that contain at least one of these balls.
>[Right Adjoint] takes a set A of balls, and tells you all the buckets that only contain balls from A.
Seems to me these intuitive explanations should be the other way around. The left adjoint seems to tell you the bucket that contains only the balls in A and right adjoint seems to tell you the buckets that contain at least one of the balls in A. Tell you the truth they both sound like they are saying the same thing to me...
What am I doing wrong here?