Are the categories, $\widehat{\mathcal{A}}\downarrow X$ and $\widehat{\mathcal{A}\downarrow X}$ equivalent. The title is fairly self explanatory. Let $\mathcal{A}$ be a small category , $X$ be an object of $\widehat{\mathcal{A}}$. Then the question is, are the categories, $\widehat{\mathcal{A}\downarrow X}$, and $\widehat{A}\downarrow X$ equivalent? I am 99% sure that this is true. Furthermore, if their is a reference for this, that would be helpful. 
 A: Actually, there is an abstract-nonsense proof (although that doesn't make constructing the equivalence by hand any less of a good exercise).  It goes through the equivalence of presheaves on a category A with discrete fibrations over A, so $\widehat{A} \simeq \mathrm{DFib}(A) \subset (\mathrm{Cat}\downarrow A)$.  Note that DFib(A) is a full subcategory of $\mathrm{Cat}\downarrow A$: any map between discrete fibrations is a map of fibrations.
Furthermore, a composite of discrete fibrations is a discrete fibration, and conversely any map between discrete fibrations is itself a discrete fibration.  Thus, for any discrete fibration B → A, we have $\mathrm{DFib}(A)\downarrow B \simeq \mathrm{DFib}(B)$.  Thus, if $B = A\downarrow X$ is the category of elements of a presheaf $X\colon A^{op} \to \mathrm{Set}$, then we have
$(\widehat{A}\downarrow X) \simeq (\mathrm{DFib}(A)\downarrow B) \simeq \mathrm{DFib}(B) \simeq \widehat{B} = \widehat{A \downarrow X}$
See also this question.
A: I'm guessing $\widehat{A}$ denotes the presheaf category? If so, see Johstone's "Sketches of an Elephant" page 8.
A: By $\widehat{\mathcal{A}}$ you must mean the category of presheaves on $\mathcal{A}$. I take it that $X$ denotes a presheaf, and that $\mathcal{A}/X$ is the comma category $y_{\mathcal{A}} \downarrow X$ where $y_{\mathcal{A}}$ is the Yoneda embedding. 
Another reference for this fact is Tom Leinster's book Higher Operads, Higher Categories, p. 394. I'm not sure who first discovered it, but it's ancient folklore. 
