Answer to **Puzzle 197**:

From [Lecture 32](https://forum.azimuthproject.org/discussion/2169/lecture-32-chapter-2-enriched-functors/p1):
>**Definition.** Let \\(\mathcal{X}\\) and \\(\mathcal{Y}\\) be \\(\mathcal{V}\\)-categories. A **\\(\mathcal{V}\\)-functor from \\(\mathcal{X}\\) to \\(\mathcal{Y}\\)**, denoted \\(F\colon\mathcal{X}\to\mathcal{Y}\\), is a function

>\[ F\colon\mathrm{Ob}(\mathcal{X})\to \mathrm{Ob}(\mathcal{Y}) \]

>such that

>\[ \mathcal{X}(x,x') \leq \mathcal{Y}(F(x),F(x')) \]

>for all \\(x,x' \in\mathrm{Ob}(\mathcal{X})\\).

Letting \\(\mathcal{X}=\mathcal{X}\\) and \\(\mathcal{Y}=\mathcal{X}\\), one has


A **\\(\mathcal{V}\\)-functor from \\(\mathcal{X}\\) to \\(\mathcal{X}\\)**, denoted \\(F\colon\mathcal{X}\to\mathcal{X}\\), is a function

\[ F\colon\mathrm{Ob}(\mathcal{X})\to \mathrm{Ob}(\mathcal{X}) \]

such that

\[ \mathcal{X}(x,x') \leq \mathcal{X}(F(x),F(x'))\]

for all \\(x,x' \in\mathrm{Ob}(\mathcal{X})\\).

Using the last definition in the Lecture above, one has

A **\\(\mathcal{V}\\)-functor from \\(\mathcal{X}^\mathrm{op}\times\mathcal{X}\\) to \\(\mathcal{V}\\)**, denoted \\(F\colon\mathcal{X}^\mathrm{op}\times\mathcal{X}\to\mathcal{V}\\), is a function

\[ F\colon\mathrm{Ob}(\mathcal{X})\times\mathrm{Ob}(\mathcal{X})\to\mathrm{Ob}(\mathcal{V}) \]

such that

\[ \mathcal{X}(x,x') \leq F(x,x')\]

for all \\(x,x' \in\mathrm{Ob}(\mathcal{X})\\).

Now letting \\(F\\) be \\(\mathrm{hom}\\),
\[\mathrm{hom}(x,x')=\mathcal{X}(x,x')\]
Satisfies the first requirement to be a \\(\mathcal{V}\\)-functor, and reduces the second requirement to:
\[\mathcal{X}(x,x')\leq\mathcal{X}(x,x')\]
which holds (as an equality).