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(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.

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Construction for a push-forward of a module over a category

This question is inspired by Example 2.3.13 in Weibel's Homological Algebra book. Perhaps it is too elementary for MO, but hopefully someone can give me a quick answer. The example is the following: …
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