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

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