I have a Pact of the Chain Warlock/Evocation Wizard character. All of my current Warlock spells are also on the Wizard spell list, except Eldritch Blast of course.
I've found a related question about a Cleric/Wizard, but that post is several years old and has some pretty conflicting answers. Not to mention it's not about Warlock and Wizard spells.
How am I able to transcribe my Warlock spells into my Wizard spellbook?
It's important to note that this question is about transcribing spells into a wizard's spellbook from other sources, not about preparing spells.
According to the "Your Spellbook" sidebar in the wizard class description:
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 spell level you can prepare and if you can spare the time to decipher and copy it.
This means that as long as a spell is on the wizard spell list and is in written form (spells don't just randomly float around after all!), it can be added to your wizard spellbook regardless of the class of the person that actually wrote the spell down. Typically, this transcription is done from some other wizard's spellbook - but not always.
Xanathar's Guide to Everything, one of several official collections of optional rules, (p. 133) lists Scribing a Spell Scroll as a downtime activity:
With time and patience, a spellcaster can transfer a spell to a scroll, creating a spell scroll.
There's a short list of what's required to create said scroll, but assuming those are met, any spellcaster using these optional rules can do it. Note that the text does not require the caster have the Spellcasting feature, merely the ability to cast a spell.
So, if any spellcaster can create a spell scroll, and if a wizard can add any spell they find to their spellbook as long as it's on the wizard spell list, the only remaining question is whether or not a warlock has access to spells on the wizard spell list. And as it turns out, most of the spells on the warlock spell list are also on the wizard spell list.
TL;DR: Assuming that the campaign is using the optional rules referenced above from Xanathar's Guide to Everything and that the warlock spell I want to add to my spellbook is also on the wizard spell list, all I have to do is scribe it onto a spell scroll and I can then transcribe it into my spellbook.
Without using optional rules, I'm afraid this is not possible. The multiclass spellcasting rules on p. 164 in the Player's Handbook explain that each class prepares spells as if it were single classed.
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. If you are a ranger 4/wizard 3, for example, you know three 1st-level ranger spells based on your levels in the ranger class. As 3rd-level wizard, you know three wizard cantrips, and your spellbook contains ten wizard spells, two of which (the two you gained when you reached 3rd level as a wizard) can be 2nd-level spells. If your Intelligence is 16, you can prepare six wizard spells from your spellbook.
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
