A Tale of Two Functors

It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it was the epoch of cohomology, it was the season of Ext, it was the season of Tor, it was the spring of short exact sequences, it was the winter of long exact sequences, we had right exactness, we had left exactness, our arrows were all going in one direction, our arrows were all reversed – in short, the period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, as the age of homological algebra.