Items for assembling artifacts cannot be sent to friends as gifts or added to your Wish List. You need artifact fixers to assemble artifacts. Fixers for creating artifacts can only be obtained by assembling collections or requesting them as gifts from your friends by adding them to your Wish List. Once you have obtained all of the items and fixers you need to assemble an artifact the "Combine" button will turn green. This means that the artifact is ready to be assembled.
The items you will get as a reward for assembling the artifact can be found right above the collection in the "Artifacts" tab.