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Apparently it's 'well known' that if $P$ is a presheaf on $C$ then there is an equivalence $\widehat{C}/P \simeq \widehat{\int P}$, where $\int P$ is the usual category of elements and $\widehat{C} = [C^{\rm op},{\rm Set}]$. (I've seen a reference to Johnstone's Topos Theory for this, but I don't have easy access to the book. It's also Exercise III.8(a) in Mac Lane--Moerdijk)

Now, I've come across comma categories $\widehat{C}/H$ for functors $H \colon D \to \widehat{C}$ (mainly when $H = D(F-,-)$ for $F \colon C \to D$), and I'd like to have a similarly useful/interesting result for them. The equivalence doesn't generalize to $\widehat{C}/H \simeq \widehat{\int H}$, or at least I can't see any way to make that work. So I want to understand the first equivalence from a more abstract-nonsensical point of view, to figure out what's really going on.

Is there a way to see the above equivalence as living in a fibrational cosmos or something similar? If not, is there any other kind of machinery that might help me understand categories of the form $\widehat{C}/H$?

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# Slices of presheaf categories

Apparently it's 'well known' that if $P$ is a presheaf on $C$ then there is an equivalence $\widehat{C}/P \simeq \widehat{\int P}$, where $\int P$ is the usual category of elements. (I've seen a reference to Johnstone's Topos Theory for this, but I don't have easy access to the book. It's also Exercise III.8(a) in Mac Lane--Moerdijk)

Now, I've come across comma categories $\widehat{C}/H$ for functors $H \colon D \to \widehat{C}$ (mainly when $H = D(F-,-)$ for $F \colon C \to D$), and I'd like to have a similarly useful/interesting result for them. The equivalence doesn't generalize to $\widehat{C}/H \simeq \widehat{\int H}$, or at least I can't see any way to make that work. So I want to understand the first equivalence from a more abstract-nonsensical point of view, to figure out what's really going on.

Is there a way to see the above equivalence as living in a fibrational cosmos or something similar? If not, is there any other kind of machinery that might help me understand categories of the form $\widehat{C}/H$?