To first give you some context, the thing on the top is from The "Representable Functors" chapter of Category Theory for Programmers. So technically, you only need to read 230 Pages of a maths textbook to get it ;)

But this isn't exactly what you asked for, so I'll try to help you get it as best I can with my limited understanding of the subject. First of all it would be helpful to know what your prior knowledge in Maths, especially Set theory, is?