Dinatural transformation,constant functor,hom functor

Question

Let $U,V$ be functors between categories $C$ and $X$ and let $Y\in Set$. Why a dinatural transformation $Y\xrightarrow{\cdot \cdot}\hom_X(U-,V-)$ is a function which assigns to $y\in Y$ a natural transformation $U\xrightarrow{\cdot} V$ , moreover with components $Uc\rightarrow Vc$ coming from dinatural transformations $U\xrightarrow{\cdot \cdot}V$?

Answer 1

There's no dinatural transformation between $F$ and $G$... however, is this an exercise coming from Mac Lane, chapter about ends and coends?

I think it's rather easy to solve it, but if you want a hint go on reading!

What should a wedge $\tau\colon Y\overset{\bullet\bullet}\to \hom(F-, G-)$ be? Isn't that condition precisely a naturality condition for $\tau(y)_c$? Now that you know this, notice that this means you have a bijective function $\{\tau_{-,y}\mid y\in Y\}\overset{\sim}\to {\rm Nat}(F,G)$, which allows us to identify the two sets. Now the diagram

where ${\rm Nat}(F,G)\to\hom(Fc,Gc)$ sends a natural transformation to its $c$-component, commutes for a single $h\colon Y\to {\rm Nat}(F,G)$, and this is precisely the desired Universal Property for $\displaystyle \int_c\hom(Fc,Gc)$.