How to get steam to recognize game files downloaded on another machine?

Yesterday I purchased GTA V for the PC via Steam and downloaded it onto my laptop at work. When I got home to transfer the 65 GB worth of files to my main computer, I cleared up 65 GB of space on my SSD and have just enough room to fit those downloaded files onto a Steam folder that I created. However, Steam won't recognize the game as installed. I tried to click the "Install" button for the game where it's currently pointed to the folder where I've copied all of this data, but after exiting Steam and opening it again, it's saying that there isn't enough room to install the game. I spent several hours looking up ways to solve this, but I haven't been able to so far.
Also, Is 65 GB enough space for this game, or will I need to clear up more room for it?
Edit: After following Ross's guide below, everything worked fine with the backup process!
Best Answer
You can do what Ross said, or if you have the WHOLE library folder with you, you can shut down steam and copy it over the on your current steam library. When you open steam, it should be there. (ps. You will need the GTA V folder ("SteamLibrary\steamapps\common\Grand Theft Auto V\") and the appmanifest_271590.acf file (SteamLibrary\steamapps\appmanifest_271590.acf), if you don't have both, steam won't recognise it.)
Note: This is a bit harder than the other solution but this doesn't require 2x size.
Note 2: I did this several times and it works.
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Quick Answer about "How to get steam to recognize game files downloaded on another machine?"
How do I get Steam to recognize installed games on a different drive?
If you had previously installed the games to an alternate location outside of your main Steam folder please try the following:Can you transfer game files to another computer Steam?
Short answer, yes! You can easily transfer Steam games to another computer, without having to redownload all the games again, using two methods actually. The one method of moving Steam games to a new PC is much faster than the other, but I will cover both in the tutorial, starting with the faster of the two.How To Fix Steam Showing Games as Uninstalled (Easy Fix)
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Answer 2
I found another way, which is to to copy the game directory on 'Steamapps' ยป 'Common'. Just start installing the game and then pause (doesn't matter when) which will make Steam create a file named "appmanifest_1234.acf" (number part will differ depending on the game) under the 'Steamapps' directory.
Open that file with a text editor and you will see there the game's name. Under that there's a line name "StateFlags" "1026". Change that number to "4", and voila! There you go, you complete the installation - simple as that.
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