Is there a universal sheaf?

Question

I highly suspect the answer to the following question is no, but this has been bugging me for a while, so I figure I might as well ask. Throughout, "site" and "continuous functor" have their meanings in SGA: a site is a category equipped with a Grothendieck topology and a functor $F:\mathcal{C}\to \mathcal{D}$ is continuous if $\mathcal{G}\circ F^{\mathrm{op}}$ is a sheaf for every sheaf $\mathcal{G}$ of sets on $\mathcal{D}$. I have two questions:

1. Let $\kappa$ be a sufficiently nice cutoff cardinal. Equip $\mathbf{Set}_{<\kappa}^{\mathrm{op}}$ with the canonical topology. If $F:\mathcal{C}\to\mathbf{Set}_{<\kappa}^{\mathrm{op}}$ is continuous, then $F^{\mathrm{op}}:\mathcal{C}^{\mathrm{op}}\to\mathbf{Set}$ is a sheaf, since the tautological presheaf $\mathbf{Set}_{<\kappa}\to\mathbf{Set}$ is representable by 1. Does the converse hold, i.e. modulo set-theoretic issues, does $\mathbf{Set}^{\mathrm{op}}$ represent the topos of sheaves, with the identity functor the universal sheaf? (Note though this may look like it has something to do with the object classifier in classifying topos theory, this is more like asking whether there is a couniversal object.)
2. More generally: if $\mathcal{D}$ is a category with the topology induced by a collection $\{\mathcal{G}_i\}$ of presheaves (SGA 4, 2.2.2) and $F:\mathcal{C}\to\mathcal{D}$ satisfies $\mathcal{G}_i\circ F^{\mathrm{op}}$ is a sheaf for every $I$, is $F$ continuous? What if $\{\mathcal{G}_i\}$ is the collection of all sheaves on $\mathcal{D}$?

The answer to the first question is positive if a sheaf $\mathbf{Set}_{<\kappa}\to\mathbf{Set}$ has to preserve arbitrary products and equalizers (with domain category's variance as I wrote it here), and I think should be easy to check is equivalent. However, while the canonical projections $2\times 2\to 2$ in $\mathbf{Set}$ generate an effective epimorphic sieve in $\mathbf{Set}_{<\kappa}^{\mathrm{op}}$, it is not universally effective epimorphic, so they are not a canonical covering of $2\times 2$, i.e. this does not directly follow from the sheaf condition.

Some of my issue is I think I have no idea how to start manufacturing sheaves on $\mathbf{Set}_{<\kappa}^{\mathrm{op}}$ to use as counterexamples besides the obvious ones.