I came across the following two constructions recently, which both have in common of relating: on the one hand the evaluation of something resembling a function, and on the other hand a more convoluted construction. My question is: are the two following constructions related?
Let $\mathcal C$ be a Category, and $G \in \hat{\mathcal{C}}$ be a presheaf over $\mathcal C$ (that is, a functor from $\mathcal C^{op}$ to $Set$). Let $x$ be in object in $\mathcal C$. Denoting by $Y(x)$ the functor $\mathcal C(\_,x) : \mathcal C^{op} \to Set$, the Yoneda Lemma tells us that the set $\hat{\mathcal C} (Y(x),G)$ is the same as the set $G(x)$.
Let $A$ be a ring, and $P \in A[X]$ be a polynomial over $A$. Let $x$ be an element of $A$. Then the image of $P$ in the quotient $A[X]/(X-x)$ is the same as $P(x)$.