Wizards cannot transcribe prepared spells from multiclassing
The "Your Spellbook" sidebar is clear that Wizards need to find a spell in a spellscroll or spellbook to transcribe it into their spellbook, then only exception given in the sidebar is for lost spellbooks:
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.
However, even a multiclassed Wizard who lost his spellbook cannot transcribe prepared spells from his other class for the following reason.
The Wizard's spellcasting states:
At 1st level, you have a spellbook containing six 1st-level wizard spells of your choice. Your spellbook is the repository of the wizard spells you know, except your cantrips, which are fixed in your mind.
According to the multiclass rules for spellcasting:
You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
Transcribing spells into a spellbook determines what spells the Wizard knows and can prepare, because the spellbook is the repository of the Wizard's known spells. Therefore, when a Wizard transcribes spells into his spellbook, he must do so as though he were not multiclassed.
However, Xanathar's Guide to Everything has rules for scribing spell scrolls as a downtime activity. If the DM gives you the opportunity, you can scribe a spell scroll for any of your multiclass spells and, if that spell is also a wizard spell, then you can copy the spell scroll into your spellbook.