Need to Show a Map is Bijective?

Of course to prove a map $\phi$ is a bijection, you can show it's one-to-one and onto. But don't forget it also suffices to produce the inverse map $\phi^{-1}$! (This holds for a general functions $f$ as well as, say, homomorphisms in algebra.) In some cases - I'm thinking group theory in particular now - it's easier to define $\phi^{-1}$ (and prove that it's a homomorphism) than to prove the injectivity and surjectivity of $\phi$. Just something to keep in mind.