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A student of mine is writing her Bachelor's thesis in which she has to use several categorical concepts (co/equalizers, co/products, subobjects, (regular) monomorphisms, etc.) in a couple of different categories (at least in the category $CMon$ of commutative monoids and in $SLatt_0$, the category of sup-semilattices with $0$). Instead of introducing these notions several times, she has decided to introduce the general notions in an abstract category and then prove that they take the form she needs in $CMon$ and in $SLatt_0$. On the other hand, now that she has these definitions, it would be nice to include examples about these constructions in other categories she knows (e.g., groups, monoids, sets, abelian groups, etc.) but, as she will not use these examples and as the draft is already long enough at this point, it would be better to avoid proofs for all these examples and just cite some references.

Hence, my question is: Can anyone suggest some reference where co/equalizers, co/products, subobjects, and similar notions are fully described, possibly with at least a hint of an argument, in several examples of commonly used categories (possibly, at least in sets, monoids, groups and abelian groups)?

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    $\begingroup$ Section IV.2 of Mac Lane's "Categories for the Working Mathematician" has a wonderful table of explicit, familiar examples of adjunctions and their unit and counit maps. Unfortunately the specific notions you asked for (co/equalizers, co/products, etc.) aren't given explicitly there, but since right adjoints preserve limits and left adjoints preserve colimits, you can infer (without much trouble) what many of those notions have to be in the given examples. This isn't quite what you asked for, so it's a comment, not an answer. Sorry if this doesn't wind up being helpful! $\endgroup$
    – user164898
    Aug 15, 2021 at 2:43
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    $\begingroup$ Lawvere & Schanuel? $\endgroup$
    – მამუკა ჯიბლაძე
    Aug 15, 2021 at 4:01
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    $\begingroup$ Adámek-Herrlich's "Abstract and concrete categories" is full of examples for every abstract concept they introduce. $\endgroup$
    – Agustí Roig
    Aug 15, 2021 at 8:28
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    $\begingroup$ Using the comment section to give answers again? ;) $\endgroup$
    – Martin Brandenburg
    Aug 15, 2021 at 9:27
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    $\begingroup$ No one mentioned Borceux's Handbook! The chapter on limits has many examples in categories of interest. There's also Awodey's book, with very explicit mentions of pushouts in Set (and with a little bit of effort, in many other categories), from which one can easily define coequalisers. $\endgroup$
    – fosco
    Aug 15, 2021 at 12:24

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