The (co)Yoneda lemma tells us that for a presheaf $F\in \hat{\mathbb{C}}$, the following formula holds: $$ F=\int^{c\in\mathbb{C}} Fc\ \times\ h_{c}\ , $$ where $h_c$ is a representable presheaf for $c$. Is there some well-known way to derive the formula for the Fourier transform of an $L^1 (G)$ function on $G^{\wedge}$ $$ \hat{f}(\chi)=\int_{G} f(x)\overline{\chi(x)}d\mu(x)\ , $$ where $\mu$ is the Haar measure on $G$, as a special case of the first formula? I'm not very knowledgeable at all on this abstract rendition of the Fourier transform, but I have been curious lately on the relationship between the Yoneda embedding and Fourier analysis. I am approaching this problem with a background in category theory, not analysis.
I really don't have motivating evidence that there is some connection between these two concepts, and I am quite aware that this may be a ridiculous hunch. Thank you for your time!