A site is subcanonical if and only if its sheafification is fully faithful?

Question

Is it true that a site is subcanonical if and only if the composite of its associated sheaf functor with the Yoneda embedding is fully faithful?

Supposing the topology is subcanonical we have the following isomorphisms: $$\mathsf{Hom}_{\widehat {\mathsf C}}(\mathbf{y}X,\mathbf{y}Y)\cong \mathsf{Hom}_{\widehat {\mathsf C}}(\iota\mathbf{ay}X,\iota\mathbf{ay}Y)\cong \mathsf{Hom}_{\mathsf{Sh}(\mathsf C,J)}(\mathbf{ay}X,\mathbf{ay}Y),$$ so $\mathbf{ay}$ is fully faithful.

Conversely however, I'm having a naturality issue. We have the chain of isomorphisms below, but only the right ones are natural. I was hoping to vary $X$ and use Yoneda to prove $\mathbf{y}Y\cong \iota\mathbf{ay}Y$ and conclude $\mathbf{y}Y$ is a sheaf... $$\mathsf{Hom}_\mathsf{C}(X,Y)\cong\mathsf{Hom}_{\widehat {\mathsf C}}(\mathbf{y}X,\mathbf{y}Y)\cong \mathsf{Hom}_{\mathsf{Sh}(\mathsf C,J)}(\mathbf{ay}X,\mathbf{ay}Y)\cong \mathsf{Hom}_{\widehat{\mathsf C}}(\mathbf{y}X,\iota\mathbf{ay}Y),$$

Applying Yoneda on the right again gives $\cong \iota\mathbf{ay}Y(X)$, but I don't see why this helps... Maybe the density of the Yoneda embedding is needed?

Added. Thinking some more, maybe there is no naturality problem. The isomorphisms $\mathbf{y}X\cong \iota \mathbf{ay}X$ are natural - they're components of the natural transformation $1\Rightarrow \iota \mathbf{a}$. Hence the left isomorphisms below are also natural, and therefore they all are. $$\mathsf{Hom}_{\widehat {\mathsf C}}(\mathbf{y}X,\mathbf{y}Y)\cong \mathsf{Hom}_{\widehat {\mathsf C}}(\iota\mathbf{ay}X,\iota\mathbf{ay}Y)\cong \mathsf{Hom}_{\mathsf{Sh}(\mathsf C,J)}(\mathbf{ay}X,\mathbf{ay}Y)$$ Finally, I think the isomorphisms $\mathsf{Hom}_\mathsf{C}(X,Y)\cong\mathsf{Hom}_{\widehat {\mathsf C}}(\mathbf{y}X,\mathbf{y}Y)$ are also natural - by the extended Yoneda lemma which says the evaluation functor $\mathsf C^\text{op}\times \widehat{\mathsf{C}}\longrightarrow \mathsf{Set}$ is naturally isomorphic to $\mathsf{Nat}(\mathbf{y}^\text{op}(-),-)$.

Without sweating through all the details, are my justifications for naturality correct? Is my attempt at a proof correct?

Added later. I think my handwaving is unjustified because even if we assume $\mathbf{ay}$ is fully faithful, we aren't given the isomorphisms. In particular, they may not be the components of the natural transformations I mentioned.

Added after giving up. I started looking for counterexamples, but then found:

Exercise 3.7.2 (Borceux, vol III). Let $(\mathsf C,J)$ be a site. Prove TFAE.

1. For all $C\in \mathsf C,R\in J(C)$, the morphisms $\mathsf C(-,f),D\in \mathsf C,f\in R(D)$ constitute in $\mathsf{PSh}(\mathsf C)$ a family of morphisms which is epimorphic and effective over the representable functors;
2. The representable functors on $\mathsf C$ are sheaves w.r.t $J$;
3. $\mathbf{ay}$ is fully faithful.

How to solve this exercise? Even better, is there a proof for $3\implies 2$ which does not use $1$?

Answer 1

You already proved that:

$$ Hom_C(X,Y) \simeq (ay Y)( X)$$

But $Hom_C(X,Y)$ is by definition $(y Y)(X)$, and all those isomorphism are functorial in $X$ so you have proved that $(y Y)( X) \simeq ay Y(X)$ which proves that $yY $ is a sheaf.

You can call that the density of the Yoneda embeddings if you want.