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There is a lot of well-known categories like Set, Grp, Ab, Top, Haus, Cpt, Met, Ring, $R$-Mod, Cat, …

Is there a well-organized list of categories like this, together with their properties like how (co)limits look like, what are the mono/epi-morphisms and so on?

Ideally, the list would also include some canonical functors between the categories, for example the forgetful functors and their adjoints. This helps with the description of limits.

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