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 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 best 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)?

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)?