Suppose that $h_{ij}(t) \geq 0$ for all $i,j$ and $t\geq 0$, and $y$ and $\tilde y$ are the outputs corresponding to inputs $u$ and $\tilde u$, respectively (with the same initial state in each case). Then $u_j(t) \geq \tilde u_j(t)$ for all $j$ and $t \geq 0$ implies $y_i(t) \geq \tilde y_i(t)$ for all $i$ and $t \geq 0$.