It’s All About Morphisms

Exercises

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., $F a = F b$ or $G a = G b$ ? (Notice that you get a cone or a co-cone this way.) Then consider cases where either $F a = G a$ or $F b = G b$ . Finally, what if you start with a morphism that loops on itself, $f : a \to a$ ?