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For $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$, let $\int X$ be its category of elements. Its objects are $(A,a)$ with $a\in X(A)$; an arrow $(A,a)\to (B,b)$ is a morphism $f:A\to B$ satisfying $X(f)b=a$. This is a small category, because $\mathcal{C}$ is small and all values of $X$ are sets.

Send $(A,a)$ to the representable presheaf $h_A$, and send an arrow $f$ to postcomposition $h_f$. There is a cocone to $X$ whose map at $(A,a)$ isThe functor laws give its naturality and the required cocone compatibility.

Compute the colimit pointwise as a disjoint union modulo its diagram identifications. At $C$ a representative is a triple $(A,a,g:C\to A)$, and it maps to $X(g)a\in X(C)$. Every $x\in X(C)$ is the image of $(C,x,1_C)$. Moreover $g$ is an arrow $(C,X(g)a)\to (A,a)$ in $\int X$, so its diagram relation identifiesThus every representative is identified with the canonical representative of its image. Two representatives have equal images exactly when their canonical representatives agree. This proves a componentwise bijection; restriction maps respect it. The resulting natural isomorphism is the canonical colimit presentation of a presheaf:

The category of elements of $X$ has objects $(W,x)$ with $x\in X(W)$ and morphisms $f:(W,x)\to (V,y)$ satisfying $X(f)y=x$. Send $(W,x)$ to the representable presheaf $H_W$ and a morphism to $H_f$. The Yoneda lemma gives a canonical cocone to $X$, whose leg at $(W,x)$ corresponds to $x$.

For any target presheaf $Z$, a cocone from this diagram to $Z$ is a family of maps $H_W\to Z$, or equivalently elements $z_{W,x}\in Z(W)$, with $z_{W,x}=Z(f)z_{V,y}$ whenever $X(f)y=x$. Such a family is exactly a natural transformation $X\to Z$, via $x\mapsto z_{W,x}$. This bijection respects the canonical cocone, so it proves its colimit universal property. Consequently the canonical colimit presentation of a presheaf isEquivalently, the density coend identifies the disjoint copies of the representables by exactly these element-category arrows. Smallness of $\mathcal{C}$ and the set-valued fibers make the indexing category small.

Write $H_c=\mathcal{C}(-,c)$. The category of elements $\int P$ of a categorical presheaf $P$ has objects $(c,x)$ with $x\in P(c)$; a morphism $(c,x)\to (d,y)$ is $u:c\to d$ satisfying $P(u)y=x$. It is a small category because $\mathcal{C}$ is small and each $P(c)$ is a set. Consider the diagram sending $(c,x)$ to $H_c$ and $u$ to the natural transformation given by postcomposition with $u$.

There is a canonical cocone ${\alpha }_{c,x}:H_c\to P$ withFunctoriality makes it a natural transformation, and the equation $P(u)y=x$ makes these transformations a cocone.

To prove its universal property, let $Q$ be any categorical presheaf with a compatible cocone ${\beta }_{c,x}:H_c\to Q$. DefineFor $u:c\to d$ and $y\in P(d)$, compatibility with the arrow $(c,P(u)y)\to (d,y)$ givesThus $t:P\to Q$ is natural. Naturality of each ${\beta }_{c,x}$ shows $t{\alpha }_{c,x}={\beta }_{c,x}$. Conversely any transformation with these equations must have the displayed values $t_c(x)$, so it is unique. We have proved the canonical colimit presentation of a presheaf:

By the canonical colimit presentation of a presheaf, writewith structure maps ${\alpha }_{c,x}:H_c\to P$. Since $P$ is a 0-presentable object, the canonical mapis bijective. Every element of a set-valued colimit has a representative in some component. In particular $1_P$ is the image of a transformation $s:P\to H_c$ for some $(c,x)$, soThus $P$ is a retract of $H_c$. The representable retract criterion from part (c), using the assumed equalisers in $\mathcal{C}$, now gives $P$ is representable. Only existence of the representative of $1_P$ is needed; no stronger uniqueness-of-factorization assertion about individual colimit injections has been assumed.

An object with the displayed property is called small projective; the term 0-presentable here uses this all-small-colimits convention. In a presheaf category, every representable presheaf has this property by the Yoneda lemma and pointwise computation of colimits. Conversely the canonical colimit presentation of a presheaf shows that the identity on such an object factors through one representable, making it a retract. If the indexing category has equalizers, a representable retract is again representable by the representable retract criterion.