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. 



Are the categories, $\widehat{\mathcal{A}}\downarrow X$ and $\widehat{\mathcal{A}\downarrow X}$ equivalent.
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. 


