If we use the definition of the Arrow category and the notation from [here](https://unapologetic.wordpress.com/2007/05/23/arrow-categories).

$$
\require{AMScd}
\begin{CD}
A @>h>> C \\
@VVfV @VVgV \\
B @>k>> D 
\end{CD}
$$

I think I can understand how the object $f$ from the $C^2$ category gets transformed using $h$ and $k$ into object $g$ if the starting category is just a directed graph (is quiver the correct name?). One just takes $h(A)$ as start and $k(B)$ as the end of the new arrow $g$.

But I'm confused when we start from the $\mathsf{Set}$ category. That is, if in our $C^2$ category the objects are functions. In particular, I don't see a way how construct the function $g\colon C \to D$ from only $f\colon A \to B,\ h\colon A \to C$ and $k\colon B \to D$? None of $f,g,h$ have $C$ as their domain, so there must be something else? Or have I misunderstood the types of $h,k$?