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A subcanonical site is one for which every representable functor is a sheaf.

For a subcanonical site $C$, the fundamental theorem of topos theory says that there is an equivalence $Sh(C/c)\cong Sh(C)/y(c)$, where $y$ is the Yoneda embedding. So the slice of a topos is the topos for a slice.

I'm wondering, can we conclude anything about whether the resulting site for $Sh(C/c)$ is subcanonical?

i.e. for each $f : b \to c$, will $P(g: a \to c) := \{ h : a \to b \mid f \circ h = g \}$ be a sheaf for the site $C/c$?

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Isn't this very basic? If $\{a_i \to b\}$ are compatible morphisms in $\mathcal{C}/c$, then these are compatible morphisms in $\mathcal{C}$, hence they glue to a unique morphism $a \to b$, and this is a morphism over $c$ since this is the case locally on $a$ and $C$ is subcanonical.

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  • $\begingroup$ It might be basic, I assumed it was basic the main references (e.g. NLab) would mention it $\endgroup$
    – Joey Eremondi
    Commented Oct 19, 2023 at 12:13
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    $\begingroup$ nlab is not a main reference for topos theory. Books are. ;) The statement is contained in Part C, Lemma 2.2.17 in Johnstone's Sketches of an elephant. The proof is basically identical to the one in my answer, but phrased in the language of sieves. $\endgroup$
    – Martin Brandenburg
    Commented Oct 19, 2023 at 18:43

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