In the following articles: https://proceedings.mlr.press/v202/yuan23b/yuan23b.pdf(section 7.4), https://arxiv.org/pdf/2207.02917.pdf(Theorem 4), The Reproducing Kernel Hilbert Spaces(RKHS) are considered as an application of the Yoneda Lemma, that interprets $K(x,y) = \langle K(x,-),K(y,-)\rangle_H$ as the Yoneda embedding. However what is the category structure here?
Here is the set up of RKHS, consider a set $X$ and $H$ a Hilbert space of real-valued functions on $X$. Under certain conditions, there exists an reproducing kernel function $K(x,-)$ in $H$ such that for any $f\in H$, $f(y) = \langle f,K(y,-)\rangle_H$ as the inner product in H. Especially when take $f=K(x,-)$, we have $K(x,y)=\langle K(x,-),K(y,-)\rangle_H$.
Now to fit this into the Yoneda embedding. We need to consider $X$ as a category and take $\text{Hom}(x,y)=K(x,y)$ as a single element set. But then how to make sense of composition of morphisms? Similarly, how to make sense of inner product as the morphism between two functions in $H$?