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).