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I first of all feel like rewriting the question, to give it a less cluttered look. Let $D$ be a finite category, and let $F: D \to C$ be a functor, where $C$ is finitely complete. Let $C^D$ denote the category of functors $D \to C$, and let $\Delta: C \to C^D$ denote the diagonal functor. Then, as usual, define the category of cones $\text{Cone}(F)$ to be the comma category

$$\text{Cone}(F) = \Delta \downarrow F.$$

A limit of $F$ is by definition a terminal object of $\text{Cone}(F)$. If $\pi: \text{Cone}(F) \to C$ is the projection functor, then the question asks whether (or under what conditions) $\lim F$ is a colimit of $\pi$.

A slightly different way of expressing the concept of limit is that the projection $\pi: \text{Cone}(F) \to C$ lifts to an equivalence

$$p: \text{Cone}(F) \simeq C/\lim F$$

which is to say that $\Sigma \circ p \cong \pi: \text{Cone}(F) \to C$, where $\Sigma: C/\lim F \to C$ is the standard projection or forgetful functor. So the question is whether $\Sigma (1_{\lim F}: \lim F \to \lim F) = \lim F$ is the colimit of $\pi$.

But notice that the colimit of the equivalence $p$ is the terminal object $1_{\lim F}$. Indeed, we may as well replace the equivalence by the identity functor, and make the key observation that the terminal object is always the colimit of the identity functor.

Now observe that $\Sigma: C/\lim F \to C$ preserves arbitrary colimits, even those over class-sized diagrams -- for example, under our hypotheses, $\Sigma$ has a right adjoint given by pulling back objects along $\lim F \to 1$, the projection to the terminal object, and left adjoints preserve arbitrary colimits. It follows that

$$\lim F \cong \Sigma(\text{colim } p) \cong \text{colim } \Sigma \circ p \cong \text{colim } \pi$$

so that the limit is indeed the colimit of the projection $\pi$.

I may as well write a little more on the "secret" significance of this sort of thing. Mac Lane in Categories for the Working Mathematician remarks (footnote, pp. 52-53) that "[comma categories] were for a time a sort of secret tool in the arsenal of knowledgeable experts". This is rather well seen by contemplating the adjoint functor theorem. The whole idea behind the adjoint functor theorem is that given a functor

$$R: C \to D$$

which is known to preserve limits, one would like to construct an initial object of the comma category $d \downarrow R$ for any object $d$ of $D$; this will be a pair $(c, d \to R c)$ which solves a universal mapping problem. How to construct the initial object? Well, it should be the limit of the identity functor, if that exists (this is dual to the key observation made above). But usually this is a large limit, whereas it is reasonable to ask only that $C$ has small limits. So the idea is to hope to find a small full subcategory inclusion $i: S \to d \downarrow R$ which is final (or cofinal, I forget which way the terminology should go), so that the limit of $i$ is the limit of the identity. (Finality means that the objects of $S$ form a weakly initial family.) So then construct the limit of $i$, and you're done. The existence of this small subcategory is called a solution set condition.

Edit: It might be clearer to some readers just to argue this directly, as follows. Let $(L, \Delta L \to F)$ denote the limit. Given a cocone $\gamma: \pi \to \Delta c$, there is a component at the limit which is just a map $L \to c$. This determines the whole cocone $\gamma$, because the value component of $\gamma$ at any other component object $(c', \Delta c' \to F)$ is a map $c' \to c$ which factors through the unique map $c' \to L$ in $\text{Cone}(F)$. In fact, any map $L \to c$ determines a cocone. Thus there is a natural bijective correspondence between maps $L \to c$ and cocones $\pi \to \Delta c$, making $L$ the colimit of $\pi$.

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Edit: It might be clearer to some readers just to argue this directly, as follows. Let $(L, \Delta L \to F)$ denote the limit. Given a cocone $\gamma: \pi \to \Delta c$, there is a component at the limit which is just a map $L \to c$. This determines the whole cocone $\gamma$, because the value at any other component $(c', \Delta c' \to F)$ is a map $c' \to c$ which factors through the unique map $c' \to L$ in $\text{Cone}(F)$. In fact, any map $L \to c$ determines a cocone. Thus there is a natural bijective correspondence between maps $L \to c$ and cocones $\pi \to \Delta c$, making $L$ the colimit of $\pi$.

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I first of all feel like rewriting the question, to give it a less cluttered look. Let $D$ be a finite category, and let $F: D \to C$ be a functor, where $C$ is finitely complete. Let $C^D$ denote the category of functors $D \to C$, and let $\Delta: C \to C^D$ denote the diagonal functor. Then, as usual, define the category of cones $\text{Cone}(F)$ to be the comma category

$$\text{Cone}(F) = \Delta \downarrow F.$$

A limit of $F$ is by definition a terminal object of $\text{Cone}(F)$. If $\pi: \text{Cone}(F) \to C$ is the projection functor, then the question asks whether (or under what conditions) $\lim F$ is a colimit of $\pi$.

A slightly different way of expressing the concept of limit is that the projection $\pi: \text{Cone}(F) \to C$ lifts to an equivalence

$$p: \text{Cone}(F) \simeq C/\lim F$$

which is to say that $\Sigma \circ p \cong \pi: \text{Cone}(F) \to C$, where $\Sigma: C/\lim F \to C$ is the standard projection or forgetful functor. So the question is whether $\Sigma (1_{\lim F}: \lim F \to \lim F) = \lim F$ is the colimit of $\pi$.

But notice that the colimit of the equivalence $p$ is the terminal object $1_{\lim F}$. Indeed, we may as well replace the equivalence by the identity functor, and make the key observation that the terminal object is always the colimit of the identity functor.

Now observe that $\Sigma: C/\lim F \to C$ preserves arbitrary colimits, even those over class-sized diagrams -- for example, under our hypotheses, $\Sigma$ has a right adjoint given by pulling back objects along $\lim F \to 1$, the projection to the terminal object, and left adjoints preserve arbitrary colimits. It follows that

$$\lim F \cong \Sigma(\text{colim } p) \cong \text{colim } \Sigma \circ p \cong \text{colim } \pi$$

so that the limit is indeed the colimit of the projection $\pi$.

I may as well write a little more on the "secret" significance of this sort of thing. Mac Lane in Categories for the Working Mathematician remarks (footnote, pp. 52-53) that "[comma categories] were for a time a sort of secret tool in the arsenal of knowledgeable experts". This is rather well seen by contemplating the adjoint functor theorem. The whole idea behind the adjoint functor theorem is that given a functor

$$R: C \to D$$

which is known to preserve limits, one would like to construct an initial object of the comma category $d \downarrow R$ for any object $d$ of $D$; this will be a pair $(c, d \to R c)$ which solves a universal mapping problem. How to construct the initial object? Well, it should be the limit of the identity functor, if that exists (this is dual to the key observation made above). But usually this is a large limit, whereas it is reasonable to ask only that $C$ has small limits. So the idea is to hope to find a small full subcategory inclusion $i: S \to d \downarrow R$ which is final (or cofinal, I forget which way the terminology should go), so that the limit of $i$ is the limit of the identity. (Finality means that the objects of $S$ form a weakly initial family.) So then construct the limit of $i$, and you're done. The existence of this small subcategory is called a solution set condition.