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If F and G are functors between two arbitrary categories C and D, does a natural transformation η from F to G always exists? What is the condition for its existence?

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Not really abstract algebra... I removed the tag – Arturo Magidin Sep 6 '10 at 21:21
up vote 4 down vote accepted

For a natural transformation $\eta$ to exist between $F$ and $G$, you need for each object $C$ of C a morphism in $D$ $\eta(C)\colon F(C)\to G(C)$. So for an easy example in which no natural transformation exists, take D to be a category with two objects, $A$ and $B$, and in which the only arrows are $1_A\colon A\to A$ and $1_B\colon B\to B$ (the two identity arrows). Take your favorite category C with at least one object, and let $F$ be the functor that maps every object of C to $A$ and every arrow of C to $1_A$, and take $G$ to be the functor that maps every object of C to $B$ and every arrow of C to $1_B$. Then there can be no natural transformation form $F$ to $G$, since there are no morphisms from $F(C)$ to $G(C)$ for any $C$ in C.

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