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)\\)