where \(X\in\mathbf{Psh}(\mathcal C)=\operatorname{Fun}(\mathcal C^{\mathrm{op}},\mathbf{Set})\) & \(\smallint X\) is the category of elements of \(X\).

From the slice to presheaves on the category of elements

Let \(p\colon Y\to X\) be an object of the slice \(\mathbf{Psh}(\mathcal C)/X\) (so \(Y\) is a presheaf and \(p\) is a natural transformation). Define a presheaf \(\widetilde Y\) on \(\smallint X\) (i.e. a functor \((\smallint X)^{\mathrm{op}}\to\mathbf{Set}\)) by

• On objects: for \((c,x\in X(c))\in\operatorname{Ob}(\smallint X)\) set \[ \widetilde Y(c,x) := \{ y\in Y(c) \mid p_c(y)=x \} = p_c^{-1}(\{x\}), \] the fiber of \(p_c\) over the element \(x\).
• On a morphism \(\alpha:(c,x)\xrightarrow{f}(d,y)\) of \(\smallint X\) (so \(f:c\to d\) in \(\mathcal C\) and \(X(f)(y)=x\)), define \[ \widetilde Y(\alpha) = Y(f)\big|_{\widetilde Y(d,y)}:\widetilde Y(d,y)\longrightarrow\widetilde Y(c,x). \] This is well-defined because for \(z\in\widetilde Y(d,y)\) we have \[ p_c(Y(f)z) = X(f)(p_d(z)) = X(f)(y)=x, \] so \(Y(f)z\in\widetilde Y(c,x)\).

Functoriality of \(\widetilde Y\) follows from functoriality of \(Y\). On morphisms \(p\mapsto\widetilde Y\) acts by sending a commutative triangle of natural transformations to the induced natural transformation of the corresponding presheaves on \(\smallint X\). This can be verified w/ similar pointwise calculations as above. Name this functor \(F:\mathbf{Psh}(\mathcal C)/X \longrightarrow \mathbf{Psh}(\smallint X)\).

From presheaves on the category of elements to the slice

Conversely, let \(A\in\mathbf{Psh}(\smallint X)\), i.e. \(A:(\smallint X)^{\mathrm{op}}\to\mathbf{Set}\). Define a presheaf \(Y_A\in\mathbf{Psh}(\mathcal C)\) together with a morphism \(p_A:Y_A\to X\) as follows.

• For \(c\in\mathcal C\) put \[ Y_A(c) := \coprod_{x\in X(c)} A(c,x). \] Write elements as pairs \((x,a)\) with \(x\in X(c)\) and \(a\in A(c,x)\).
• For \(f:c\to d\) in \(\mathcal C\) define \[ Y_A(f)\colon Y_A(d)=\coprod_{y\in X(d)}A(d,y)\longrightarrow \coprod_{x\in X(c)}A(c,x)=Y_A(c) \] on a summand element \((y,a)\in A(d,y)\) by \[ Y_A(f)(y,a) = \bigl(X(f)(y), A\bigl(f:(c,X(f)(y))\to(d,y)\bigr)(a)\bigr). \] Here \(f\) determines a morphism \((c,X(f)(y))\to(d,y)\) in \(\smallint X\), and \(A\) (viewed contravariantly) gives a map \(A(d,y)\to A(c,X(f)(y))\). This defines \(Y_A(f)\), and straightforward checks give the functoriality \(Y_A(f\circ g)=Y_A(g)\circ Y_A(f)\).
• Define \(p_A(c)\colon Y_A(c)\to X(c)\) by projection \((x,a)\mapsto x\). Naturality of \(p_A\) follows from the construction: \[ X(f)\bigl(p_A(d)(y,a)\bigr)=X(f)(y)=p_A(c)(Y_A(f)(y,a)). \]

Thus \((Y_A,p_A)\) is an object of the slice. On morphisms \(A\mapsto (Y_A\to X)\) is defined in the obvious coordinatewise way, giving a functor \(G:\mathbf{Psh}(\smallint X)\longrightarrow \mathbf{Psh}(\mathcal C)/X\).

The two constructions are quasi-inverse

We check \(G\circ F\cong\mathrm{id}\) and \(F\circ G\cong\mathrm{id}\) naturally.

• Start with \((Y\overset p\to X)\). Applying \(F\) gives the fiber presheaf \(\widetilde Y\) on \(\smallint X\). Applying \(G\) to \(\widetilde Y\) produces \[ Y_{\widetilde Y}(c)=\coprod_{x\in X(c)}\widetilde Y(c,x)=\coprod_{x\in X(c)} p_c^{-1}(\{x\}). \] But \(\coprod_{x} p_c^{-1}(\{x\})\) is canonically in bijection with \(Y(c)\) (every element \(y\in Y(c)\) sits in the unique fibre over \(p_c(y)\)). These bijections are natural in \(c\) and respect restriction maps, so we get a canonical isomorphism of presheaves \(Y_{\widetilde Y}\cong Y\) which commutes with the projections to \(X\). Hence \(G(F(Y\to X))\cong (Y\to X)\) naturally.
• Conversely, start with \(A\in\mathbf{Psh}(\smallint X)\). Applying \(G\) gives \(Y_A=\coprod_{x\in X(-)} A(-,x)\) with projection \(p_A\). Applying \(F\) we get the fiber presheaf \(\widetilde{Y_A}\) on \(\smallint X\) with \[ \widetilde{Y_A}(c,x)= \{ (x',a)\in Y_A(c)\mid p_A(c)(x',a)=x \}. \] But only those summands with \(x'=x\) contribute, so \(\widetilde{Y_A}(c,x)\cong A(c,x)\) canonically. These identifications are natural and compatible with the action on morphisms in \(\smallint X\), hence \(F(G(A))\cong A\) naturally.

Thus \(F\) and \(G\) are inverse equivalences of categories.

Conclusion

We have constructed explicit mutually inverse functors that proves

hence the slice category \(\mathbf{Psh}(\mathcal C)/X\) is (canonically) equivalent to \(\mathbf{Psh}(\smallint X)\) ■

Slightly different & higher level proof.

Let \(\mathcal C\) be a small category, \(X\in\widehat{\mathcal C}=\mathbf{Psh}(\mathcal C)=\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathbf{Set})\). Let \(\smallint X\) denote the category of elements of \(X\). Write \(y:\mathcal C\to\widehat{\mathcal C}\) for the Yoneda embedding; an element \(x\in X(c)\) corresponds (by Yoneda) to a morphism \(x: y(c)\to X\) in \(\widehat{\mathcal C}\).

We will produce canonical mutually quasi-inverse functors

and show they are inverse.

Define \(F\) by representables / Yoneda

For an object \(p:Y\overset{p}{\to}X\) of the slice \(\widehat{\mathcal C}/X\) define a presheaf \(F(p)\) on \(\smallint X\) by the rule

That is: at the object \((c,x)\in\smallint X\) we take the set of morphisms in the slice from the representable map \(x:y(c)\to X\) into \(p\). Functoriality in \((c,x)\) is given by precomposition in the slice; this is tautologically functorial.

This definition is purely representable: for fixed \((c,x)\) the functor

is representable (represented by the object \(y(c)\xrightarrow{x}X\) itself). Thus \(F(p)\) is just the functor of slice-maps from representables, packaged as a presheaf on \(\smallint X\).

Define \(G\) by left Kan extension / colimit (canonical reconstruction)

Conversely, let \(A\in\widehat{(\smallint X)}\). Compose the projection functor \(\pi:\smallint X\to\mathcal C\) with \(y\) to get representables indexed by \(\smallint X\):

There is a canonical coend / coproduct (colimit) presentation

(where \(A(c,x)\cdot y(c)\) denotes the coproduct of copies of the representable \(y(c)\) indexed by the set \(A(c,x)\)). The colimit carries a canonical map \(p_A:Y_A\to X\) induced from the maps \(y(c)\xrightarrow{x}X\) used in the colimit diagram. This produces an object \(p_A:Y_A\to X\) of the slice \(\widehat{\mathcal C}/X\). Define \(G(A):= (Y_A\to X)\).

Concretely this is the left Kan extension (along \(\pi\)) of \(A\) followed by the canonical map into \(X\); nothing ad hoc — just the universal colimit reconstructing a presheaf on \(\mathcal C\) from its fibres over representables.

\(F\) and \(G\) are quasi-inverse (Yoneda / representability check)

Now we'll check the two composites are naturally isomorphic.

• For \(p:Y\to X\) in the slice, consider \(G(F(p))\). By construction \(G(F(p))\) is the colimit of copies of the representables \(y(c)\) indexed by the sets \[ F(p)(c,x)=\operatorname{Hom}_{/X}(y(c)\xrightarrow{x}X,\;Y\to X) \] with the structure maps given by the defining maps \(y(c)\xrightarrow{x}X\). Universal property of that colimit exactly yields the canonical map from the colimit to \(Y\) which is an isomorphism: indeed the identity of \(Y\) corresponds uniquely to the cocone of maps from each representable-summand into \(Y\). Thus there is a canonical isomorphism in \(\widehat{\mathcal C}/X\) \[ G(F(p)) \cong p \] This is the usual expression of a presheaf as the colimit of representables weighted by its values on representables (the indexed Yoneda colimit).
• For \(A\in\widehat{(\smallint X)}\), consider \(F(G(A))\). By definition, for each \((c,x)\in\smallint X\), \[ F(G(A))(c,x)=\operatorname{Hom}_{/X}\bigl(y(c)\xrightarrow{x}X,\;Y_A\xrightarrow{p_A}X\bigr) \] But maps from the representable \(y(c)\xrightarrow{x}X\) into the colimit \(Y_A\) are in canonical bijection with elements of \(A(c,x)\) (those are precisely the generators of the colimit), and naturality in \((c,x)\) identifies the whole presheaf \(F(G(A))\) with \(A\). Again this bijection is the representable/cocone universal property so we get a canonical isomorphism \[ F(G(A)) \cong A \]

These isomorphisms are natural, hence \(F\) and \(G\) are mutually quasi-inverse equivalences.

Conclusion

The functor

is an equivalence of categories with inverse given by the left Kan extension/colimit construction \(G\). All verifications reduce to the universal (representability/Yoneda) properties of the representables and of colimits.

Hence

as required. ■