Let $\mathfrak{C}$ be a small category endowed with the trivial Grothendieck topology $J$. I must show that $Sh(\mathfrak{C},J)=[\mathfrak{C}^{op},{\bf Set}]$. I take an object $C$ and a matching family for $M_C$ (the only element of $J(C)$. It is easy to verify the existence of an amalgamation. It is the element $x_{1_C}$, which is the image by the matching family of $1_C:C \longrightarrow C$. Now, I must prove uniqueness of the amalgamation. Assume by contradiction the existence of $y$ such that $P(f)(y)=x_f$ for each $f \in M_C$. I cannot understand how to find a contradiction and, hence, to verify that $x_{1_C}$ agrees with $y$.
Here, there is an answer:
Presheaves are sheaves for the trivial topology
However, I need a proof using my argument. Can you help me, please?
If such a $y$ exists, then since $1_C$ is in $M_C,$ you have $y = P(1_C)(y)=x_{1_c}$, where the second equality is by assumption.