What are the objects of Hom$(y,x)$ in the contra-variant Yoneda functor I am reading about the contra-variant Yoneda functor and I am bit confused about what objects actually are in the Hom set. More specifically, if $Hom(-,x):\mathcal{C}^{op}\rightarrow Set$ is the contra-variant Yoneda functor, then are the elements of $Hom(-,x)(y) = Hom(y,x)$ morphisms in $\mathcal{C}$ between $y$ and $x$? Or are they morphisms in $\mathcal{C}^{op}$ from $y$ to $x$, which then would correspond to morphism $x$ to $y$ in $\mathcal{C}$?

The confusion comes from the notation $C^{op}$, which is often used only to indicate that the functor is contravariant.

The covariant Yoneda lemma uses the covariant functor $Hom(x,-):C\to Set$, while the contravariant Yoneda lemma uses the _contravariant_ functor $Hom(-,x):C\to Set$. This contravariant functor is equivalent to the covariant functor $Hom(x,-):C^{op}\to Set$ if we want to be strict with the notation, but as I said, usually one only means by $C^{op}\to Set$ that the functor is contravariant.

To sum up, $Hom(-,x)(y)=Hom(y,x)$ is the set of morphism $y\to x$ in $C$, which is the same as morphisms $x\to y$ in $C^{op}$.