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Given a topos $\mathcal{T}$, and an internal site $\mathbb{C} \in Cat(\mathcal{T})$, there should be a nice way to describe the topos of internal sheaves on $\mathbb{C}$ in terms of morphisms in $\hat Cat(\mathcal{T})$ (the 2-category of large categories in $\mathcal{T}$, being some sub-2-category of 2-sheaves in $Func(\mathcal{T}^op, \hat Cat)$) from $\mathbb{C}^{op}$ to the codomain fibration / self-indexing $\mathbb{T}$ of $\mathcal{T}$, i.e. regarding an internal sheaf as a morphism $\mathbb{C}^{op} \to \mathbb{T}$ in $\hat Cat(\mathcal{T})$.
This should be obvious. But is there a nice discussion from this point of view somewhere in the literature?
B2.3.13?
Ah! :-)
Okay, so I was looking for such a pointer to include in the new entry internal sheaf. But I’ll announce this in another thread… here.
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