Answer by Simon Henry for Stone duality for the algebra of Boolean functions...
I might be missing something, but I think you are overcomplicating things.I clain that your topos classifies the theory $T$ of pairs $(B,\phi)$ where $B$ is a boolean algebra and $\phi : B \to \{0,1\}$...
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$\newcommand\FinSet{\mathit{FinSet}}\newcommand\FinBool{\mathit{FinBool}}\newcommand\FreeFinBool{\mathit{FreeFinBool}}\newcommand\Set{\mathit{Set}}\newcommand\Psh{\mathit{Psh}}$It's well-known that the...
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