TITLE: The conceptual difference in notations of Cat
QUESTION [5 upvotes]: There have been some places in which a conceptual difference (that of criteria of identification, etc) is avoided in the categorical notation, namely

Let $Cat$ be at the same time the 2-category of small categories and its subjacent 1-category i.e. the category of small categories (...)

Why one can do this? The 2-category CAT and the 1-category Cat are not remarkably different? What is Cat (generally) in the literature..?

REPLY [7 votes]: Here I take terminology fromnLab: 
As long as 2-category means strict 2-category this usage is exactly the same as using $M$ to name both a manifold and its set of points.
If $Cat$ is meant to be some weak 2-category then you need to specify what enrichment you want--or as you put it what criterion of identification.  There is not only one possibility.  In that case the usage only makes sense if one specified enrichment is being assumed in the context.