TITLE: Terminology in category theory
QUESTION [28 upvotes]: A lot of the terminology of category theory has obvious antecedents in analysis: limits, completeness, adjunctions, continuous (functors), to name but a few.  However, analysis and category theory seem to be at opposite poles of the spectrum.
Is there anything deep here, or is it a case of "it has wings, so let's call it a duck"?
This was partly inspired by the top-rated answer to the question What is a metric space? and by the (slightly unsatisfactory answers to the) question Can adjoint linear transformations be naturally realized as adjoint functors?.  In particular, in the first case - the categorical view of metric spaces - there does seem to be an obvious route between the two worlds, do the terminologies correspond there?

REPLY [10 votes]: As for the limit example, it is rooted in analysis by way of topology.
A topology on some set X is a system of subsets, partially ordered by inclusion, that have arbitrary joins and finite meets (or possibly the other way around depending on whether you're looking for open or closed sets), with X and 0 members of the system.
This partial order gets us a category structure, consisting of open (closed?) sets. Limits of these turn out to be interpretable as limits in the classical sense; and colimits are the dual concept as usual.