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,

Does a category have all finite 2-limits if it has all 2-equalizers and 2-products?

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.