The Yoneda embedding of the walking-arrow category computed objectwise

The hom-sets are $\mathcal{W}(0,0)=\{1_0\}$, $\mathcal{W}(0,1)=\{u\}$, $\mathcal{W}(1,1)=\{1_1\}$, and $\mathcal{W}(1,0)=\mathrm{\varnothing }$. These are exactly the four object values in the displayed table.

A presheaf represented by $a$ acts on $u^{\mathrm{o}\mathrm{p}}:1\to 0$ by precomposition with $u$. For $a=0$ this is the empty function $\mathrm{\varnothing }\to \{1_0\}$; for $a=1$ it sends $1_1$ to $1_1\circ u=u$ by [F1].

By [F2], $y(u)$ is postcomposition with $u$. At $0$ it sends $1_0$ to $u\circ 1_0=u$, and at $1$ it is the unique empty function. This proves the asserted component table, including the empty hom-set.

There is one natural transformation $y(0)\Rightarrow y(0)$ and one $y(0)\Rightarrow y(1)$, namely the identity and $y(u)$. There is no transformation $y(1)\Rightarrow y(0)$ because its component at $1$ would be a function $\{1_1\}\to \mathrm{\varnothing }$, forbidden by [F3].

A transformation $y(1)\Rightarrow y(1)$ has forced singleton-to-singleton components, and the naturality square commutes by [F1], so it is the identity. Hence the four natural-transformation sets have the same empty-or-singleton table as the four hom-sets, exactly as [L1] asserts.

Composition in the image has only the identity composites and $y(u)$ composed with an identity; by [F1] and step 2.2 these reproduce the composition of $1_0,u,1_1$ in $\mathcal{W}$.