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In chapter 2.5, excercise #7 of Mac Lane's Categories for the working mathematician, the question reads,

If $T:A\rightarrow D$ is a functor, show that its arrow functions $T_{a,b}:A(a,b)\rightarrow D(Ta,Tb)$ define a natural transformation between functors $A^\mathrm{op}\times A\rightarrow \mathbf{Set}$.

I am stuck because I am not sure what the question means. If possible, I would like a clarification and be pointed in the right direction. Thanks!

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1 Answer 1

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Here's a starting point: the two functors $F,G:A^{op}\times A\to \mathsf{Set}$ we want to consider are $$F(a,b)=\text{Hom}_A(a,b)$$ $$G(a,b)=\text{Hom}_D(Ta,Tb)$$ For two $(a,b),(c,d)\in A^{op}\times A$, we have that $$\text{Hom}_{A^{op}\times A}((a,b),(c,d))=\text{Hom}_{A^{op}}(a,c)\times\text{Hom}_A(b,d)=\text{Hom}_A(c,a)\times\text{Hom}_A(b,d)$$ Do you see how the two functors act on morphisms? We need to be able to take a morphism from $(a,b)$ to $(c,d)$ in $A^{op}\times A$, i.e. an ordered pair $(f,g)\in \text{Hom}_A(c,a)\times\text{Hom}_A(b,d)$, and create a corresponding morphism $F(f,g)$ from $F(a,b)=\text{Hom}_A(a,b)$ to $F(c,d)=\text{Hom}_A(c,d)$, and similarly with $G$ replacing $F$. It may help to write down the information we start with: $$(f,g)\in\text{Hom}_{A^{op}\times A}((a,b),(c,d))\qquad\text{ i.e. }\quad f:c\to a, \;\; g:b\to d$$ and using this information, we want to define some $F(f,g)\in\text{Hom}_{\mathsf{Set}}(F(a,b),F(c,d))=$ $\text{Hom}_{\mathsf{Set}}(\text{Hom}_A(a,b),\text{Hom}_A(c,d))$. So, $F(f,g)$ should input morphisms $p:a\to b$ and output morphisms $q:c\to d$. (Use $f$ and $g$ to do this.)

The "arrow functions" of the functor $T:A\to D$ are just, for each pair of objects $a,b$ of $A$, the function $T_{(a,b)}$ from $\text{Hom}_A(a,b)$ to $\text{Hom}_D(Ta,Tb)$ that sends $f:a\to b$ to $Tf:Ta\to Tb$ (recall that a functor needs to both send objects to objects and morphisms to morphisms).

A natural transformation between $F$ and $G$ would consist of a collection of morphisms (in $\mathsf{Set}$) $\eta_{(a,b)}:F(a,b)\to G(a,b)$, one for each $(a,b)\in A^{op}\times A$, satisfying $$\eta_{(c,d)}\circ F(f,g)= G(f,g)\circ\eta_{(a,b)}$$ for every $(a,b),(c,d)\in A^{op}\times A$ and $(f,g)\in\text{Hom}_{A^{op}\times A}((a,b),(c,d))$. So, this problem wants you to show that the collection of maps $T_{(a,b)}:F(a,b)\to G(a,b)$ do in fact satisfy this property (but in order to do that, you need to figure out what $F(f,g)$ and $G(f,g)$ are, which I gave a hint on above).

(By the way, in case it was not clear, I am using the notation $\text{Hom}_C(x,y)$ instead of the notation $C(x,y)$ to refer to the set of morphisms from $x$ to $y$, for $x,y\in\text{Ob}(C)$, where $C$ is a category.)

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