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?