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
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>$F\colon\mathrm{Ob}(\mathcal{A})\to \mathrm{Ob}(\mathcal{B})$
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>such that
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>$\mathcal{A}(a, a') \leq \mathcal{B}(F(a),F(a'))$
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>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.