Suppose $C$ and $D$ are sites and $F$, $G:C\to D$ two functors connected by a natural transformation $\eta_c:F(c)\to G(c)$.
Suppose further that two functors $\hat F$, $\hat G:\hat C\to\hat D$ on the respective categories of presheaves are given by $\hat F(c)=F(c)$ and $\hat G(c)=G(c)$ where I abuse the notation for the Yoneda embedding.
Is there always a natural transformation $\hat\eta_X:\hat F(X)\to \hat G(X)$?
The problem is, that in the diagram $ \begin{array}{rcccccl} \hat F(X)&=&\operatorname{colim} F(X_j)&\to& \operatorname{colim} G(X_j)&=&\hat G(X)\\ &&\downarrow &&\downarrow\\ \hat F(Y)&=&\operatorname{colim} F(Y_k)&\to& \operatorname{colim} G(Y_k)&=&\hat G(Y) \end{array} $ for a presheaf morphism $X\to Y$ the diagrams for the colimits may be different, or am I wrong?