Construction of Yoneda extension

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 $\mathbb{Set}^{ \mathbb{C}^{op} } \rightarrow \mathcal{E}$.