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Let $\mathbf{C}$ denote a category and suppose $X$ is an object of $\mathbf{C}$. Then intuitively, there should be a "forgetful something" from the hom-functor $\mathrm{Hom}_\mathbf{C}(-,X)$ to the slice-category $\mathbf{C}/X.$ But its just not clear to me what this forgetful something is. Since the domain is a functor, my first guess would be that it is a "forgetful natural transformation". But since the functors $\mathrm{Hom}_\mathbf{C}(-,X) : \mathbf{C} \rightarrow \mathbf{Set}$ and $\mathrm{id}_{\mathbf{C}/X} : {\mathbf{C}/X} \rightarrow {\mathbf{C}/X}$ aren't parallel, there is no concept of a "natural tranformation" from $\mathrm{Hom}_\mathbf{C}(-,X)$ to $\mathbf{C}/X.$

So, what kind of a thing is the forgetful thing $\mathrm{Hom}_\mathbf{C}(-,X) \rightarrow \mathbf{C}/X$?

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  • $\begingroup$ I don't think that's very intuitive... $\operatorname{Hom}_{\mathbf C}(-, X)$ is essentially the same thing as $X$, because of the Yoneda lemma (it's part of the subcategory of $\operatorname{Fun}(\mathbf{C}^{op}, \mathbf{Set})$ of representable functors, which is equivalent to $\mathbf{C}$). I don't really see what sort of "transformation" $X \to \mathbf{C}/X$ could be meaningful. And I don't really see what kind of information you could "forget", too... $\endgroup$ – Najib Idrissi Jan 27 '15 at 16:20
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    $\begingroup$ You can see $\operatorname{Hom}(A,X)$ as the fiber over $A$ of the forgetful functor $\mathbf{C}/X \to \mathbf{C}$. I think that makes more sense. Maybe some notion of fibration can work here (Grothendieck fibration, Cartesian fibration -- I'm not really an expert). $\endgroup$ – Najib Idrissi Jan 27 '15 at 16:23
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    $\begingroup$ I'm not sure it is what you're looking for but one link between the two is the following : $\hom(-,X)$ is the colimit of the diagram $\mathbf C/X \to \mathbf C \to [\mathbf C^{\mathrm{op}},\mathsf{Set}]$ (where the first arrow is the forgetful functor and the second the Yoneda embedding). Another way too see it : $\mathbf C/X$ is the category of elements of $\hom(-,X)$. $\endgroup$ – Pece Jan 27 '15 at 18:31
  • $\begingroup$ @NajibIdrissi The forgetful functor $\mathcal{C}_{/ X} \to \mathcal{C}$ is precisely the discrete fibration corresponding to $\mathcal{C} (-, X)$, via the Grothendieck construction. $\endgroup$ – Zhen Lin Jan 27 '15 at 22:29
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For a functor $F : \mathcal{C} \to \mathsf{Set}$ you may consider its category of elements $\int F$. The objects are pairs $(X,s)$ with $X \in \mathrm{Ob}(\mathcal{C})$ and $s \in F(X)$. It is clear how to define the morphisms.

We have $\int \mathrm{Hom}_{\mathcal{C}}(-,X) \cong \mathcal{C}/X$ and $\int \mathrm{Hom}_{\mathcal{C}}(X,-) \cong X/\mathcal{C}$.

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