I was reading "Categories for the Working Mathematician" by Saunders Mac Lane and on page 28 I was puzzled by a question. Hom-sets were being explained and the author leaves it to the reader to describe a natural transformation (for functors $S,T:\mathbf C\to \mathbf B$) $\eta:S\to T$ in terms of functions $\mathbf C(c,c')\to \mathbf B(Sc,Tc')$ and I can't figure it out.
I apologise if this has been asked before but I couldn't find anything like this.
Best Answer
Yes, this is a bit tricky for page 28! (General tip: this is the most difficult introduction to category theory that's every been published, and most students are well served by having another reference at hand.)
The original definition of $\eta$ gives, for each $c\in\mathbf C$, a morphism $\eta_c:Sc\to Tc$. Now, for each morphism $f:c\to c'$, we have a commutative square witnessing the naturality condition $T(f)\circ \eta_c=\eta_{c'}\circ S(f)$. Thus there is a well defined morphism $\eta_f: S(c)\to T(c')$ given by either of the equal routes around this square. Then one can rephrase the naturality condition as $$\eta_f=T(f)\circ \eta_{\mathrm{id}_c}=\eta_{\mathrm{id}_{c'}}\circ S(f).$$ For a related alternative, one can view $\eta$ as a functor $\mathbf{C}\to \mathbf{D}^{\bullet\to\bullet}$, where the codomain is the category whose objects are arrows of $\mathbf{D}$ and whose morphisms are commutative squares. Then the action on morphisms is essentially $f\mapsto \eta_f$ as described above.