Category Theory for Programmers Chapter 17: It's All About Morphisms

  1. Consider some degenerate cases of a naturality condition and draw the appropriate diagrams. For instance, what happens if either functor $F$ or $G$ map both objects $a$ and $b$ (the ends of $f:a \to b$) to the same object, e.g., $Fa = Fb$ or $Ga = Gb$? (Notice that you get a cone or a co-cone this way.) Then consider cases where either $Fa = Ga$ or $Fb = Gb$. Finally, what if you start with a morphism that loops on itself — $f:a \to a$.