It's not hard to see that a category is finitely complete if it has finite products and equalizers. In short, this is because one can write all limits as iterations of these two "operations". 


I wonder if there is a 2-version of this. In particular,



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> Does a category have all finite 2-limits if it has all 2-equalizers and 2-products?
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My instinct is no, and that we will need another(or several more) limits to build all 2-limits. 


Of course the question can be generalized to n-limits, and I'm curious about that also. 



2-completeness analog of completeness theorem
Your suspicion is correct: in general, a V-category has all weighted V-limits if it has all conical V-limits *and* is cotensored over V (see Kelly's *Basic Concepts of Enriched Category Theory*, section 3.10). For V = Cat (and this is true for bicategories too), cotensors can be constructed from conical limits and cotensors with the arrow category 2, although I don't know the original reference for that. 'Finite' limits are a bit more complicated in the enriched case, but see Street's 'Limits indexed by category-valued 2-functors', JPAA 1972, for the 2-case.


