$\def\C{\underline C}\def\hom{\mathbf{Hom}}\def\ker{\mathbf{Ker}}$ Let $\C$ be a category, let $C$ be a sieve on $\C$ with a basis $\{S_i\}$ and let $F:C^{op}\to\mathbf{Set}$ be a functor.
Then Topologies et Faisceaux by Demazure (Proposition 4.1.4, p.198) states that
There is a functorial isomorphism in F: $$ \hom(C,F)\cong\ker\left(\prod_i\hom(S_i,F)\rightrightarrows\prod_{i,j}\hom(S_i\times S_j,F)\right). $$
But I cannot find this isomorphism.
I think for any $f\in\hom(C,F)$, we can send it to $\prod_i(f_i)$ where $f_i$ sends elements of a non-empty $\hom(X,S_i)$ to the unique element in the image of $f_X:C(X)\rightarrow F(X)$, and this belongs to the kernel.
I am confused why this is an isomorphism. I know not how to send an element in the kernel to $\hom(C,F)$.
Any hint or reference is sincerely welcomed.