David, you've mucked up your substitution into the definition. What you've written is not right. Let me change the names of the things in the definition. That might make things less confusing.

>**Definition.** Let \\(\mathcal{A}\\) and \\(\mathcal{B}\\) be \\(\mathcal{V}\\)-categories. A **\\(\mathcal{V}\\)-functor from \\(\mathcal{A}\\) to \\(\mathcal{B}\\)**, denoted \\(F\colon\mathcal{A}\to\mathcal{B}\\), is a function

>

>\[ F\colon\mathrm{Ob}(\mathcal{A})\to \mathrm{Ob}(\mathcal{B}) \]

>

>such that

>

>\[ \mathcal{A}(a, a') \leq \mathcal{B}(F(a),F(a')) \]

>

>for all \\(a,a' \in\mathrm{Ob}(\mathcal{A})\\).

You need to check that this holds when \\(\mathcal{A}\\) is \\(\mathcal{X}^\mathrm{op}\times\mathcal{X}\\), when \\(\mathcal{B}\\) is \\(\mathcal{V}\\) and when \\(F\\) is \\(\text{hom}\\). You'll find you need to use the structures of \\(\mathcal{X}\times \mathcal{Y}\\) and \\(\mathcal{X}^{\mathrm{op}}\\) given by John in the lecture.

>**Definition.** Let \\(\mathcal{A}\\) and \\(\mathcal{B}\\) be \\(\mathcal{V}\\)-categories. A **\\(\mathcal{V}\\)-functor from \\(\mathcal{A}\\) to \\(\mathcal{B}\\)**, denoted \\(F\colon\mathcal{A}\to\mathcal{B}\\), is a function

>

>\[ F\colon\mathrm{Ob}(\mathcal{A})\to \mathrm{Ob}(\mathcal{B}) \]

>

>such that

>

>\[ \mathcal{A}(a, a') \leq \mathcal{B}(F(a),F(a')) \]

>

>for all \\(a,a' \in\mathrm{Ob}(\mathcal{A})\\).

You need to check that this holds when \\(\mathcal{A}\\) is \\(\mathcal{X}^\mathrm{op}\times\mathcal{X}\\), when \\(\mathcal{B}\\) is \\(\mathcal{V}\\) and when \\(F\\) is \\(\text{hom}\\). You'll find you need to use the structures of \\(\mathcal{X}\times \mathcal{Y}\\) and \\(\mathcal{X}^{\mathrm{op}}\\) given by John in the lecture.