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Is there a general construction or existence theorem for the fibered product of two subcategories of some ambient category? What sort of problems might one run into? Does this require a 2-categorical construction?

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The category of small categories is monadic over $[\bullet \rightrightarrows \bullet, \textbf{Set}]$, hence in particular has all small limits. But the real question is – does this necessarily give you what you want? The fibred product of two subcategories of a fixed category is just their intersection, as in any concrete category. –  Zhen Lin Aug 20 '12 at 1:59
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