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?](https://mathoverflow.net/questions/5957/what-is-a-metric-space) and by the (slightly unsatisfactory answers to the) question [Can adjoint linear transformations be naturally realized as adjoint functors?](https://mathoverflow.net/questions/476/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?