Showing that the sheaf-functor $\epsilon: \tilde{\sf C} \to \tilde{\tilde{\sf C}}$ is an equivalence

For simplicity, assume $(\mathbb{C}, J)$ is a small subcanonical site. The quasi-inverse of the embedding $\mathbf{Sh}(\mathbb{C}, J) \to \mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J))$ has a very simple description: it is the functor that sends a sheaf $F : \mathbf{Sh}(\mathbb{C}, J)^\mathrm{op} \to \mathbf{Set}$ to its restriction along the embedding $\mathbb{C} \to \mathbf{Sh}(\mathbb{C}, J)$.

Indeed, suppose $F : \mathbf{Sh}(\mathbb{C}, J)^\mathrm{op} \to \mathbf{Set}$ is a sheaf. I claim $F$ is determined up to unique isomorphism by its restriction along the embedding $\mathbb{C} \to \mathbf{Sh}(\mathbb{C}, J)$. Indeed, let $X : \mathbb{C}^\mathrm{op} \to \mathbf{Set}$ be a $J$-sheaf. Then $X$ is the colimit of a canonical small diagram of representable sheaves on $(\mathbb{C}, J)$ in a canonical way. Consider the colimiting cocone on $X$: it is a universal effective epimorphic family and is therefore a covering family in the canonical topology on $\mathbf{Sh}(\mathbb{C}, J)$. Thus, $F (X)$ is indeed determined up to unique isomorphism by the restriction of $F$ to $\mathbb{C}$. We must also show that the restriction is actually a sheaf on $(\mathbb{C}, J)$; but this is true because $J$-covering sieves in $\mathbb{C}$ become universal effective epimorphic families in $\mathbf{Sh}(\mathbb{C}, J)$.

Thus we obtain a functor $\mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J)) \to \mathbf{Sh}(\mathbb{C}, J)$ that is left quasi-inverse to the embedding $\mathbf{Sh}(\mathbb{C}, J) \to \mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J))$, and the argument above shows that it is also a right quasi-inverse.