$\def\sfC{\mathsf{C}} \def\op{\mathrm{op}} \def\set{\mathsf{Set}} \def\psh{\operatorname{PSh}} \def\ob{\operatorname{Ob}} \def\hom{\operatorname{Hom}}$Let $\sfC$ be a category. A presheaf over $\sfC$ is a functor $\sfC^\op\to\set$. Denote $\psh(\sfC)$ to the category of presheaves on $\sfC$. Denote $Y:\sfC\to\psh(\sfC)$ to the covariant Yoneda embedding. Let $c$ be an object in $\sfC$. There is a natural equivalence of categories $\psh(\sfC/c)\simeq \psh(\sfC)/Yc$ (see Proposition 3.4 here).
I'm interested in finding a mention in the literature or on the internet of the following result:
Let $F:\sfC^\op/c\to\set$ be a presheaf and let $\eta:F'\Rightarrow Yc$ be the associated natural transformation of presheaves on $\sfC$ via the previous equivalence. Then $F$ is representable if and only if $F'$ is representable.
The proof goes like this: The functor $F'$ maps $d\in\ob(\sfC)$ to $\coprod_{f \in \sfC(d,c)} F(f)$. Then one can verify that $F$ is represented by $(d\to c)\in\ob(C/c)$ with universal element $\xi\in Fd$ if and only if $F'$ is represented by $d\in\ob(\sfC)$ with universal element $(d\to c,\xi)\in F'd$. That is, if we consider the natural transformations $(\sfC/c)(-,d)\Rightarrow F$ induced by $\xi$ and $\sfC(-,d)\Rightarrow F'$ induced by $(d\to c,\xi)$, then one must prove that they both are isos iff at least one of them is an iso.
The reason why I want to find it explained somewhere is because I stumbled upon it on my own and I'm interested in seeing if there is any kind of remark or theory around worth of mention.