The category of atomic monoids with atom-preserving homomorphisms is complete and cocomplete, with explicit product, coproduct, equalizer, and pullback constructions and length-set formulas.
Borceux, Handbook of categorical algebra 1 , Cambridge University Press, 1994
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The Category of Atomic Monoids: Universal Constructions and Arithmetic Properties
The category of atomic monoids with atom-preserving homomorphisms is complete and cocomplete, with explicit product, coproduct, equalizer, and pullback constructions and length-set formulas.