# The Unapologetic Mathematician

## An awful lot of natural maps

There are a bunch of natural (in both senses) maps we can consider now. Some look all but tautological over $\mathbf{Set}$, and we may have used them in the past without comment. In the enriched context, though, we should go over them.

1. The family of arrows $\eta_C:\mathbf{1}\rightarrow\hom_\mathcal{D}(S(C),T(C))$ is extraordinarily $\mathcal{V}$-natural exactly when $\eta_C:S(C)\rightarrow T(C)$ is $\mathcal{V}$-natural.
2. For a $\mathcal{V}$-functor $T:\mathcal{C}\rightarrow\mathcal{D}$, the map $T_{A,B}:\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{D}(T(A),T(B))$ is natural in both $A$ and $B$.
3. Similarly, if $T:\mathcal{A}\otimes\mathcal{B}\rightarrow\mathcal{C}$ is a $\mathcal{V}$-functor, then $T(\underline{\hphantom{X}},1_B):\hom_\mathcal{A}(A,A')\rightarrow\hom_\mathcal{C}(T(A,B),T(A',B))$ is $\mathcal{V}$-natural. And it’s also natural in $B)$.
4. In particular, $\hom_\mathcal{C}(C,\underline{\hphantom{X}})_{A,B}:\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(C,B)^{\hom_\mathcal{C}(C,A)}$ is natural in all three variables.
5. Putting together naturalities 1 and 4 tells us that for a morphism $f:\mathbf{1}\rightarrow\hom_\mathcal{C}(A,B)$, the transformation $\hom_\mathcal{C}(1_C,f):\hom_\mathcal{C}(C,A)\rightarrow\hom_\mathcal{C}(C,B)$ is natural.
6. The “evaluation” $e:Z^Y\otimes Y\rightarrow Z$ is natural in both variables.
7. Since we built compositions from evaluations and the closure adjunction, the arrow $\hom_\mathcal{C}(B,C)\otimes\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(A,C)$ is natural in all variables.
8. The identity functor on $\mathcal{C}$ has an identity natural transformation, so by naturality 1 we see that $i_C:\mathbf{1}\rightarrow\hom_\mathcal{C}(C,C)$ is natural.
9. All the monoidal structural isomorphisms — $\alpha_{A,B,C}:(A\otimes B)\otimes C)\rightarrow A\otimes(B\otimes C)$, $\lambda_A:\mathbf{1}\otimes A\rightarrow A$, $\rho:A\otimes\mathbf{1}\rightarrow A$, and $\gamma_{A,B}:A\otimes B\rightarrow B\otimes A$ — are natural in all variables.
10. We can start with $(Z^Y)^X$ and hit it with $\underline{\hphantom{X}}\otimes Y$ to get $(Z^Y\otimes Y)^{X\otimes Y}$. Then we can evaluate to get $Z^{X\otimes Y}$. This is an isomorphism we call $p_{X,Y,Z}^{-1}$, corresponding to the adjunction $\hom_\mathcal{C}(A\hom_\mathcal{C}(B,C))\cong\hom_\mathcal{C}(A\otimes B,C)$, and it’s natural in all variables.
11. We can compose the following arrows:
• $\lambda_X^{-1}:X\rightarrow\mathbf{1}\otimes X$
• $j_{X\otimes Y}\otimes1_X:\mathbf{1}\otimes X\rightarrow(X\otimes Y)^{X\otimes Y}\otimes X$
• $p_{X,Y,X\otimes Y}\otimes1_X:(X\otimes Y)^{X\otimes Y}\otimes X\rightarrow((X\otimes Y)^Y)^X\otimes X$
• $e_{X,(X\otimes Y)^Y}:((X\otimes Y)^Y)^X\otimes X\rightarrow(X\otimes Y)^Y$

and we get the “coevaluation” $d_{X,Y}$ — the counit of the closure adjunction. And thus the coevaluation is $\mathcal{V}$-natural in all variables.

12. We can compose the coevaluation $d_{X,\mathbf{1}}:X\rightarrow(X\otimes\mathbf{1})^\mathbf{1}$, and then use the right unit isomorphism to get a $\mathcal{V}$-natural isomorphism $X\rightarrow X^\mathbf{1}$.
13. In general, a family $f:T(D,D,A,B)\otimes S(E,E,A,C)\rightarrow R(F,F,B,C)$ is $\mathcal{V}$-natural in any of its variables if the corresponding variables are natural in $\overline{f}:T(D,D,A,B)\rightarrow R(F,F,B,C)^{S(E,E,A,C)}$

Whew. That’s a mouthful. It can be instructive to sit down and try to interpret some of these in the context of categories enriched over $\mathbf{Set}$, so when you recover I’d advise taking a look at that.