Let's say you add $\ce{HCl}$ to water. The $\ce{H+}$ ion concentration increases and that causes a decrease in $\mathrm{pH}$. But why would the $\mathrm{pOH}$ increase?
I can't see why added $\ce{H+}$ ions will decrease the $\ce{OH}$$\ce{OH-}$ concentration? Is this because of the water auto-ionization? Is it because, according to Le Chatelier principle, when you add an amount of a component, the reaction will go in the other direction to counteract the disturbance, thus decreasing the concentration of $\ce{OH}$$\ce{OH-}$ to form water?