How to download Steam games purchased off the iPhone app?

I purchased a game last night through the phone app and went on Steam this morning on my computer and cant seem to find the purchased game. Under activity it says that Ive bought it but I cant seem to figure out where or how I download. Can someone please help? This is my first time buying a game through the phone.

I'im pretty sure you have to click the pull-down menu in Steam that usually says "installed" (should be in the upper right hand corner of Steam) and change it to "all games" or something like that. Then all the games you have purchased should be listed, including the one you bought through the Steam app.

Hover over (or click) the big Library text - should get a menu.

Yeah, always make sure the thing to the right of "Search" under library says "All Games," that way you can see games you purchased but haven't downloaded yet.

The other posters are all correct.

If it's not showing in your library I suggest restarting Steam. Sometimes new items that weren't purchased through the client need that to show up in the library. You can also always go to the game's store page in the client and launch/install it from there if you don't want to restart it.