general rule here that's worth noting.

> **Proposition**: Given \\(\Phi: X \nrightarrow Y, \Psi: Y \nrightarrow Z, \Omega: X \nrightarrow Z\\), suppose \\(\forall x \in \text{Ob}(X)\\) and \\(\forall z \in \text{Ob}(Z)\\) we have

> \[ \forall y \in \text{Ob}(Y) \quad \Phi(x, y) \otimes \Psi(y, z) \leq \Omega(x, z) \]

> \[ \exists y \in \text{Ob}(Y) \quad \Phi(x, y) \otimes \Psi(y, z) \geq \Omega(x, z) \]

> Then \\(\Psi\circ\Phi = \Omega\\).

The first condition tells us that the join \\(\bigvee\big(\Phi(x, y) \otimes \Psi(y, z)\big) \leq \Omega(x, z)\\).

The second condition tells us that the join \\(\bigvee\big(\Phi(x, y) \otimes \Psi(y, z)\big) \geq \Omega(x, z)\\).

(because a join is a *least* upper bound and a least *upper bound* respectively!) QED