Theorem 1. section 7 of chapter III(page 76), of Maclane's categories for working mathematicians, says any functor $K:\mathcal{D}\rightarrow Set$ can be represented as the colimit of a certain diagram of representable functors. I am a bit confused about this certain diagram, which in Maclane proof is given by $$J ^ \mathcal{D}\rightarrow Set^\mathcal{D}$$ where $J$ is the category of elements of $K$ (or $1 \downarrow K)$. What mkes me confused is that this functor is defined by $\left(d,x\right) \mapsto x$, where $d$ is an object of $\mathcal{D}$ and $x:* \rightarrow Kx$.

First, I would like to ask what is this diagram of representable functors which can recover the $K$.

Secondly, I would like to know what is the dual of this theorem (which supposedly establish an equivalence of any contravariant set valued functor with a limit of certain diagram of representable functors)

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    $\begingroup$ You dualised the statement incorrectly. The correct dualisation is obtained by replacing $\mathcal{D}$ with $\mathcal{D}^\mathrm{op}$. Unfortunately that means everything is still in terms of colimits. $\endgroup$ – Zhen Lin Jan 15 '14 at 7:44

There is a typo in Maclane. Not $J^D$, but $J$, or, more precisely, $J^{op}$(or we have to say that $M$ is contravariant).

Let $D$ be a locally small category, $K\colon D\to\mathbf{Set}$ be a functor. Category of elements of $K$ is the comma category $(*\downarrow K)$. There exists a natural projection: $$ pr\colon(*\downarrow K)\to D, $$ which sends every pair $(d,x)\in(*\downarrow K)$ to the object $d\in D$. You can also consider it's dual: $$ pr^{op}\colon(*\downarrow K)^{op}\to D^{op}. $$ Another functor we need - Yoneda functor: $$ Y\colon D^{op}\to\mathbf{Set}^{D}, $$ which sends every object $d\in D$ to the representable functor $hom_D(d,-)$. Now we can take the composition: $$ (*\downarrow K)^{op}\xrightarrow{pr^{op}}D^{op}\xrightarrow{Y}\mathbf{Set}^{D}. $$ Thus, we have the functor $M=Y\circ pr^{op}$: $$ (*\downarrow K)^{op}\xrightarrow{M}\mathbf{Set}^D. $$ There exists a cocone $\varphi\colon M\to\Delta_K$, such that $\varphi(d,x)=\alpha_x$, where $\alpha_x$ is the image of $x$ by the Yoneda-lemma-isomorphism $K(d)\cong Nat(hom_D(d,-),K)$. It is an exercise to check that cocone $\varphi$ is limiting, and, consequently, $K$ is the colimit of $M$.

We can see that $M(d,x)$ is representable functor for any $(d,x)\in(*\downarrow K)$. Thus, functor $K$ is a colimit of a "diagram of representable functors".

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    $\begingroup$ It is interesting to note that this theorem is only present in the second edition of CWM. In the first edition there is no trace of it, as far as I can see. $\endgroup$ – magma Jan 17 '14 at 0:45

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