trying the discover function equation mapping using inner set probe idea, need help

[ALine]

I have been thinking about how you could probe within a map or function in algebra and decided to try to use a sort of "probe" set in-between the sets that are being mapped together. It reminds me of composition and am trying it from that angle, however I need some help with it. The reason I want to do this is because I want to see if I can "discover" a mapping without the need of a mapping(for higher order math's) or function equation for algebra right now.

 

The name could use some work for the topic.