I think we can tidy up that last bit @Matthew.

For all \$$c\in\text{Ob}(\mathcal{A})\$$ we have \$$\mathcal{A}(a, c)\otimes\Phi(c, b) \leq \bigvee\mathcal{A}(a, x)\otimes\Phi(x, b)\$$

Set \$$c = a\$$ and we get \$$\mathcal{A}(a, a)\otimes\Phi(a, b) \leq \bigvee\mathcal{A}(a, x)\otimes\Phi(x, b)\$$

Then \$$I\leq\mathcal{A}(a, a)\$$ tells us \$$\Phi(a, b) \leq \bigvee\mathcal{A}(a, x)\otimes\Phi(x, b)\$$