Construction of Yoneda extension In "Category Theory" by Steve Awodey, there is a suggestion for the reader to construct a left adjoint in the proof of the UMP of the Yoneda embedding. Namely, for any small category $\mathbb{C}$, the Yoneda embedding has the UMP in the sense that for any cocomplete category $\mathcal{E}$ and a functor $F:\mathbb{C} \rightarrow \mathcal{E}$, there is a (unique up to natural isomorphism) colimit preserving functor $F_{!}:\mathbb{Set}^{ \mathbb{C}^{op} } \rightarrow \mathcal{E}$ s. t. $F_{!} \circ y \cong F $ as indicated by the following diagram:


The proof starts with taking some functor $P$ in $\mathbb{Set}^{ \mathbb{C}^{op} }$ and writing it as a colimit of representable functors:
$$ P \cong \lim_{ \rightarrow_{i \in \mathbb{I}} }  yC_{i}, $$
where the index category is the category of elements:
$$  \mathbb{I} = \int_{ \mathbb{C} } P.$$
The functor $F_{!}$ acts on objects as follows:
$$  F_{!}(P) = \lim_{ \rightarrow_{i \in \mathbb{I}} } F(C_{i}). $$
Question: how does it act on arrows?
Update 1:
This question
Kan extensions for linear categories 
mentions the definition of the Kan extension (an instance of which the Yoneda extension is) taking arrows into consideration as well. It defines the required functor via colimit inclusions, but the method does not look like "direct" or "explicit" construction as it only introduces a condition on arrows. Also, induced inclusions are not completely clear to me, so the definition seems to be circular.
 A: The trick is to compare the arrows in $\mathbb{C}$ to those in $\mathbb{Set}^{ \mathbb{C}^{op} }$ or, if you prefer these terms, compare arrows in $\mathbb{C}$ to natural transformations in $\mathbb{Set}^{ \mathbb{C}^{op} }$.  You can express this using homsets in the style of your answer but I will express it by diagrams.  It would be worth working through explicitly for some small finite categories $\mathbb{C}$. I recommend actually drawing the diagrams.
You have $h:P\rightarrow Q$ in $\mathbb{Set}^{ \mathbb{C}^{op} }$, with $P$ and $Q$ colimits for specified diagrams in $\mathbb{Set}^{ \mathbb{C}^{op} }$.  Those are Yoneda images of diagrams in $\mathbb{C}$ so you know how to map those diagrams into $\mathcal{E}$ via $F:\mathbb{C} \rightarrow \mathcal{E}$.  Then, as you do in your post you define $F_!$ as mapping colimits to colimits to define $F_!(P)$.  
Now the desired $F_!(h)$ is an $\mathcal{E}$ arrow from the colimit of one diagram, to the colimit of another, which by the colimit property means a cone in $\mathcal{E}$ from the first diagram to the colimit of the second.  
A vertex of the first cone corresponds to an arrow $k:y(C_i)\rightarrow P$, so $hk:y(C_i)\rightarrow Q$ corresponds to a vertex of the diagram over $F_!(Q)$ and it has a colimit injection (not necessarily monic) to $F_!(Q)$.  Simple verification shows these injections commute with the diagram arrows over $F_!(Q)$ and so form a cone from the diagram over $F_!(P)$ to $F_!(Q)$.  So they induce an arrow $F_!(h):F_!(P)\rightarrow F_!(Q)$.  Trivially, composing this arrow with the one induced by a further $\mathbb{Set}^{ \mathbb{C}^{op} }$ arrow  $j:Q\rightarrow R$ is just the same as inducing an arrow by $jh:P\rightarrow R$.  It is functorial $\mathbb{Set}^{ \mathbb{C}^{op} } \rightarrow \mathcal{E}$.
A: It seems to me the answer is quite simple. On objects $P: C^{op} \to Set$ we have $F_!(P) = \int^{c \in C} P(c) \cdot F(c)$, where for a set $X$, the notation $X \cdot c$ indicates a coproduct of an $X$-indexed collection of copies of $c$. Let $\theta: P \to Q$ be a natural transformation. Then 
$$F_!(\theta) := \int^c P(c) \cdot F(c) \stackrel{\int^c \theta(c) \cdot F(c)}{\to} \int^c Q(c) \cdot F(c).$$ 
Let me write it a little differently, without using coend notation. We have a coequalizer diagram 
$$\sum_{c, c'} P(c') \cdot \hom(c, c') \cdot F(c) \rightrightarrows \sum_c P(c) \cdot F(c) \to F_!(P)$$ 
where one of the parallel arrows uses the covariant action of $C$ on $F$ and the other the contravariant action of $C$ on $P$. Thus, using the universal property of the coequalizer $F_!(P)$, the arrow $F_!(\theta): F_!(P) \to F_!(Q)$ is the unique one making the following diagram commute: 
$$\begin{array}{ccccc}
\sum_{c, c'} P(c') \cdot \hom(c, c') \cdot F(c) & \rightrightarrows & \sum_c P(c) \cdot F(c) & \to & F_!(P) \\ 
^{\sum_{c, c'} \theta(c') \cdot id \cdot id} \downarrow & & ^{\sum_c \theta(c) \cdot id} \downarrow  & & \downarrow \\
\sum_{c, c'} Q(c') \cdot \hom(c, c') \cdot F(c) & \rightrightarrows & \sum_c Q(c) \cdot F(c) & \to & F_!(Q)
\end{array}
$$
where we use the fact that the left and center vertical arrows make the two squares on the left commute serially (this uses naturality of $\theta$ for one of those squares). 
