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The question is in the title.

My current research subject is the homotopy theory of $2$-categories. It may be slightly unreasonable to expect my PhD thesis to be read or even looked at by people outside the field of higher categories, but for some reasons I would like this text to be enough self-contained so as to allow someone without any prior knowledge of $2$-category theory to understand at least the basic definitions. "Someone without any prior knowledge of $2$-category theory" does not refer to someone not knowing anything about category theory. However, I do not want to assume the reader to know anything more advanced than the very basic definitions of category theory. I therefore would like to be able to point out the reader to a text which does not require more advances prerequisites than elementary category theory. Otherwise I shall write the full definitions myself if no such text is available. There are some texts introducing $2$-categories in the literature, but all the references I have come across are ruled out because of the following requirements:

It really should contain the unravelled definitions of $2$-category, lax functor and lax transformation;

I do not want the word "enriched" to be used;

If the word "natural" is used, then the meaning of the naturality should be explained in full;

The vocabulary has to be consistent with current usage;

The $2$-categories have to be strict because I really do not make any use of any other objects. This condition is, however, less important than the previous ones, but it would be clumsy to write things like "for the unravelled definitions, see Leinster's "Basic Bicategories" and strictify the coherence conditions".

I think the "big.list" tag is not approppriate because I am really looking for a single reference which would fulfill these conditions.

EDIT: Thanks to those people who have commented on this question. Unfortunately, I think that none of the references provided gives what I was looking for. The closest I have found is Tom Leinster's "Basic Bicategories" which I had cited in the question, the rub being that it is an introduction to general bicategories, which implies coherence diagrams much more complicated than in the context of $2$-categories. The way to simplify them in this special case soon becomes obvious once one has acquired some experience, but I do not want to impose this task on a reader whom I shall assume without any prior acquaintance with $2$-categorical notions, let alone bicategories. The logical conclusion is that I will start from Tom's exposition and adapt it to the framework of $2$-categories. Many thanks for having written that text, Tom!

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I won't give this as an answer because I can't confirm your conditions, but at least it discusses these topics and can be searched: math.tamu.edu/~maguiar/a.pdf –  Eric Rowell Nov 25 '11 at 22:25
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+1. I would like 2-categories for the working mathematician. The books I find on 2-categories and bicategories assume much more of a category theory background than one would get from reading Mac Lane's classic. Given how people doing locally compact groupoids in analysis are using bicategories all the time as is Morita theory for rings, it be nice to have a treatise for noncategory theorists. –  Benjamin Steinberg Nov 27 '11 at 18:44
    
the better I read was "J. W. Gray, Formal category theory--Adjointness for 2 categories" Is hard in first time, but is great book. –  Buschi Sergio Dec 25 '11 at 11:54
    
Unfortunately, Gray's book fails the condition "vocabulary has to be consistent with current usage" in a massive way. (-: –  Mike Shulman Jan 2 '12 at 18:08
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As far as I can tell, both Borceaux's Handbook and Johnstone's Topos theory book have the definitions you want. –  S. Carnahan Jan 9 '12 at 14:12
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