Justifying a Natural Isomorphism Claim Involving Limits and Colimits From Categories for the Working Mathematician pg. 68-69:


Question: What in particular justifies the natural isomorphism claims of (2) and (3)? What are the full functors that the natural isomorphism is from?
 A: Developing a little bit on Derek Elkins'comment above, for each small category $\mathbf C$ you can consider the functor 
$$P=\text{Nat}[F,\Delta-] \colon \mathbf C \longrightarrow \mathbf {Set}$$ 
that sends each object $c \in \mathbf C_0$ in $P(c)=\text{Nat}[F,\Delta(c)]$ and each morphism $f \in \mathbf C_1$ in the function $P(f)=\text{Nat}[F,\Delta(f)]$.
You can observe that a colimit object per $F$ is nothing but a representing object for such functor. Indeed a colimit is given by:


*

*an object $\varinjlim F \in \mathbf C_0$ 

*a cone from $\varinjlim F$ to $F$, that is a $\tau \in \text{Nat}[F,\Delta(\varinjlim F)]=P(\varinjlim F)$


such that for each other pair $(c \in \mathbf C, \sigma \in P(c)=\text{Nat}[F,\Delta(c)])$ there is a unique $f \colon c \to \varinjlim F \in \mathbf C_1$ satisfying the equation
$$\sigma = \Delta(f) \circ \tau=\text{Nat}[F,\Delta(f)](\tau)=P(f)(\tau)\ .$$
This amounts to saying that $(\varinjlim F,\tau)$ is a universal object for $P$. By yoneda lemma we have family of isomorphisms
$$\text{Nat}[\mathbf C[c,-],P] \cong P(c)$$
natural in $c \in \mathbf C_0$ and $P \in [\mathbf C,\mathbf{Set}]$, these isomorphism associate to the colimit object $(\varinjlim F,\tau)$ the natural transformation 
$$\mathbf C[\varinjlim F,-] \longrightarrow P=\text{Nat}[F,\Delta-]$$
that to each $f \colon \varinjlim F \to c$ associates the object $P(f)(\tau)$.
The universal property for colimits states that this natural transformation is actually a biiection, hence it is a natural isomorphism and is the natural isomorphism you are looking for.
The converse also holds, that is every natural isomorphism of the form $\hom[c,-] \to P$ arise from a pair $(c,\sigma \in P(c))$ such that $(c,\sigma)$ is a colimit for $F$.
By duality the same (or to be correct the opposite) holds for limits too.
Hope this helps.
A: Here is a more simplistic approach: I'll consider (3) only. We have the diagram $\left \{ F_i \right \}_{i\in \Lambda}$ and a colimit cocone $(l,\lambda).$
For the moment, fix $c$. Then the claim is that there an isomorphism of sets,
$\phi_c:Nat(F,\Delta c)\to C(l, c)$. 
But this is clear because  $Nat(F,\Delta c)$ is just the set of cocones from $F$ to $c$ and the definition of the colimit says precisely that there is a bijection between this set and the set of arrows $f: l\to c,\ $ defined as follows: $\phi_c$ maps the cocone $(c,\tau)$ to the unique $f:l\to c$ that satisfies $f\circ \lambda_i=\tau_i.$ Thus, $Nat(F,\Delta c)\cong C$(lim$F,c).$ 
Naturality follows by considering, for $f:c\to c'$, the square 
\begin{array}{&&} 
Nat(F,\Delta c) & \stackrel{\phi_c}{\to}& C(\text{lim F},c) \\ 
\downarrow & & \downarrow  \\ 
 Nat(F,\Delta c')& \stackrel{\phi_{c'}}{\to} & C(\text{lim F},c')
\end{array}
which is seen to commute as soon as we apply the UMP of the colimit. 
