Internal sets in a category Let $\mathcal{C}$ be any category with finite limits, and write $S$ for the category of finite sets, thought of as an essentially algebraic theory.
Do we always recover $\mathcal{C}$ itself when considering Set-models in $\mathcal{C}$? (i.e. finite-limit preserving functors $S^{op}$ to $\mathcal{C}$)?
 A: Yes indeed; with a view towards Fosco's comment, product-preserving functors $S^{op} \to C$ automatically preserve finite limits. This is related to Gabriel-Ulmer duality, but we can argue without that apparatus. 
First, I claim it suffices to check this in the case $C = \text{Set}$. This is because the Yoneda embedding $y: C \to \text{Set}^{C^{op}}$ both preserves and reflects (finite) limits, so that $F: S^{op} \to C$ preserves finite limits iff $y \circ F: S^{op} \to \text{Set}^{C^{op}}$ does. But since finite limits in $\text{Set}^{C^{op}}$ are computed pointwise, it suffices to check that for all $c \in \text{Ob}(C)$, the composite 
$$S^{op} \stackrel{y F}{\to} \text{Set}^{C^{op}} \stackrel{\text{eval}_c}{\to} \text{Set}$$ 
preserves finite limits if it preserves finite products; this establishes the claim. 
Now to finish, a product-preserving functor $F: S^{op} \to \text{Set}$ is determined up to (unique) isomorphism by the value $X = F(1)$. If $i: S \to \text{Set}$ is the full inclusion, this $F$ is given by $\text{Set}(i-, X)$ (since $\text{Set}(i-, X)$ is after all product-preserving and returns the value $X$ at the argument $1$). But since $i$ preserves finite colimits, the functor 
$$Set(i-, X) = (S^{op} \stackrel{i^{op}}{\to} \text{Set}^{op} \stackrel{Set(-, X)}{\to} \text{Set})$$ 
preserves finite limits. 
