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Let $U,V$ be functors between categories $C$ and $X$.
Why a dinatural transformation $\{*\}\xrightarrow{\cdot \cdot}\hom_X(U-,V-)$ is a function which selects a natural transformation $U\xrightarrow{\cdot} V$ ?
Essentially by definition.
Are you asking for somebody to unwind the definition a bit more?
Urs, the OP asked similar questions on math.SE, which I promptly (and hopefully properly) answered. The main problem was that he (she?) misunderstood the meaning of the construction proposed by Mac Lane. I hope now all has been clarified.
If not, dear Jan Pax, feel free to contact me! (just comment on the thread on math.SE)
Okay, thanks!
If there is any explanation that would usefully be added to the nLab entry dinatural transformation, please consider doing so.
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