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Or would they have to go through the process, and cost, of scribing a scroll of, say Detect Magic, then pass an Arcana check to 'learn' and spend more precious gp to scribe it into their spellbook?
If the Artificer multiclassed into Wizard, could they then prepare said spells and directly Scribe them? Thereby skipping the scroll creation to learning process.
How spellbooks work is explained on p. 114 PHB:
Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it. (...) You can only copy wizard spells into the book, no matter if the spell is prepared, from a scroll, or from another spell book of yours. Not all Artificer spells are also Wizard spells, so you would not be able to copy cure wounds or aid.
In addition, the Spellcasting section under Multiclassing (PHB p. 164) says:
Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
So even if you have access to a spell that it also on the Wizard list via your Artificer levels, it is not a Wizard spell for you, it is an Artificer spell, and you cannot copy it into your book. Again, this is idependent of if you have prepared the spell or not.
The rules for copying prepared spells are a bit indirect (also on P 114, PHB):
Replacing the Book. You can copy a spell from your own spellbook into another book—for example, if you want to make a backup copy of your spellbook. This is just like copying a new spell into your spellbook, but faster and easier, since you understand your own notation and already know how to cast the spell. You need spend only 1 hour and 10 gp for each level of the copied spell.
If you lose your spellbook, you can use the same procedure to transcribe the spells that you have prepared into a new spellbook. Filling out the remainder of your spellbook requires you to find new spells to do so, as normal. For this reason, many wizards keep backup spellbooks in a safe place.
It's generally understood that there is nothing stopping you from making such a copy from your prepared spells, even if you have not yet lost your old spellbook.
