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If I know that a category $\mathcal{D}$ is equivalent to a category of sheaves $\mathbf{Sh}(\mathcal{C},J)$, and I fix the small category $\mathcal{C}$, does the Grothendieck topology $J$ have to be unique?

(I know that there can be different sites of definition of the Grothendieck topos $\mathcal{D}$, but I am not interested in that. I only want to know how much (if at all) $J$ can vary if somehow I know what $\mathcal{C}$ is.)

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It does depend on what structure you are implicitly respecting. For example, if $\mathcal{C}$ is a subcategory, and you insist that the composite $\mathcal{C} \hookrightarrow \mathcal{D} \xrightarrow{\equiv} \mathbf{Sh}(C, J)$ is naturally isomorphic to the Yoneda embedding, then $J$ is indeed unique, since this determines a specific geometric embedding $\mathcal{D} \hookrightarrow \mathbf{PSh}(\mathcal{C})$, and equivalence classes of such are bijective with Grothendieck topologies on $\mathcal{C}$.

But if you really care about bare equivalences of categories, then no; $\mathbf{PSh}(\mathbf{1} + \mathbf{1})$ has two different subtoposes that are equivalent to $\mathbf{Set}$, and they correspond to different topologies on $\mathbf{1} + \mathbf{1}$.

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    So, no two categories of sheaves for distinct topologies on the same base can ever be equivalent? – 2017-01-17
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    @Kevin: Well, okay yes, but there's more structure here. There are two different topologies on $\mathbf{1} + \mathbf{1}$ that make $\mathbf{Sh}(\mathbf{1} + \mathbf{1}, J) \equiv \mathbf{Set}$, but the two corresponding functors $\mathbf{1} + \mathbf{1} \to \mathbf{Set}$ are not isomorphic. – 2017-01-17
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    @Hurkyl thanks. What do you mean by "it depends on what structure"? In what I have in mind, $F:\mathcal{C}\to \mathcal{D}$ is full and faithful - $\mathcal{C}$ is a full subcategory of $\mathcal{D}$. – 2017-01-17
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    @Johnny: I've restated the first paragraph to hopefully make it clear. – 2017-01-17
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    Thanks again. Well, I want $\mathcal{D}$ to be a formal cocompletion of $\mathcal{C}$, so what you are saying implies that yes, $J$ is unique. Is this fact written somewhere in this clarified way? – 2017-01-17
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    @JohnnyGreen: I was looking at [this nlab page](https://ncatlab.org/nlab/show/dense+sub-site) as a reference to try and straighten out the details. Note that the yoneda embedding gives an equivalence $\mathcal{D} \equiv \mathbf{Sh}(\mathcal{D}, \text{canonical topology})$, so $(\mathcal{D}, \text{canonical topology})$ is the "large site" in theorem $2.4$. – 2017-01-17