Suppose $\mathcal{C}_i$ are sites, $\mathcal{C}$ is a category, and $\mathcal{F}_i:\mathcal{C}_i\rightarrow\mathcal{C}$ are functors. Assign to $\mathcal{C}$ the smallest topology such that all $\mathcal{F}_i$ are continuous (see, e.g., SGA 4, Exposé III, Proposition 3.6). Does it hold that $\mathcal{G}:\mathcal{C}^{\mathrm{op}}\rightarrow\mathbf{Set}$ is a sheaf if and only if every $\mathcal{G}\circ\mathcal{F}_i$ is? This would have nice consequences, e.g. a topology on $\mathbf{Set}^{\mathrm{op}}$ such that $\mathrm{Id}$ is the universal sheaf, describing sheaves completely internally on the category of sites. No reference seems to consider this question, which makes me doubtful.
I'm following the conventions+definitions in SGA and ignoring set-theoretic issues.