Let $C$ be a site (in my case with coverings only isomorphisms and with finite products, but I think that's not necessary here) $X$ an object of $C$ such that $Hom(U,X)$ is non-empty for every $U$ in $C$. Let $F$ be a sheaf on $C$. I want to show that $Hom_{PSh(C)}(\star,F)\tilde{=}F(X)$ functorial in $F$, where $\star$ denotes the one-point presheaf.
I have 3 ideas that I'm stuck at for basically the same reason.
Idea 1: Construct $\Psi: Hom_{PSh(C)}(\star,F)\to F(X),\ f\mapsto f_X(\star)$
However, I can't show surjectivity, because for $U\in C$ there may be two morphisms $\pi,\pi^\prime:U\to X$, so setting $f_U(\star)$ as the restriction to $U$ of an $a\in F(X)$ doesn't quite work.
Idea 2: By Yoneda, we have that $F(X)\tilde{=}Hom(h_X,F)$ and the condition on the morphism into X means that $\phi:h_X\to\star$ is surjective as presheaves. By this we get $\Phi^*:Hom_{PSh(C)}(\star,F)\to F(X)$ is injective. But surjectivity gives me the same problems for basically the same reason.
Idea 3: We have that $F(X)=Hom_{PSh(C/X)}(\star,F)$ but for the same reasons as above I fail to show that this is isomorphic to $Hom_{PSh(C)}(\star,F)$
I would be very grateful if you could give me a hint how this can be proven or where I am wrong. Thank you in advance.
Your claim is simply not true in general. We can stick to presheaves to find a counter example. Assume that $C$ has a terminal object. The one-point presheaf is $y(\mathbf 1)$ because $C(x,\mathbf 1)$ is a one-element set for each $x\in C_0$. Here $y$ denotes the Yoneda-embedding. You have that $$Hom(y(\mathbf 1),P) = P(\mathbf 1)$$ functorial in $P$ because of the Yoneda-embedding. Let me can an object $c$ such that $C(x,c)$ is non-empty for each $x$ weakly terminal. If your statement is true, then we get that $P(c) = P(\mathbf 1)$ for every weakly terminal object $c$ and presheaf $P$. For example when the base site is the category $C = Fin$ of all non-empty finite sets, then every object in $C$ is weakly terminal! Can you construct a presheaf on $Fin$ that is not constant?