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