Construction of pullbacks in the category of Monoids (book reference)

Question

Can someone give me a book reference where I can find the construction of the pullbacks (or of one pullback) in the category of Monoids?

Answer 1

See this book at page 412. It might be useful https://books.google.ro/books?id=fXgFCAAAQBAJ&pg=PA412&lpg=PA412&dq=pullbacks+of+two+monoid+homomorphisms&source=bl&ots=8x1RnVULeU&sig=6ixihj4MegdRUqpm8HXQN5LcIjU&hl=en&sa=X&ved=0ahUKEwiE8-fw0ezTAhUFbBoKHX6QBskQ6AEIOjAD#v=onepage&q=pullbacks%20of%20two%20monoid%20homomorphisms&f=false

Answer 2

If $A \to C \leftarrow B$ are monoid homomorphisms, then the underlying set of the pullback $A \times_C B$ is $|A \times_C B| = |A| \times_{|C|} |B|$, i.e. the pullback of the underlying sets, and the multiplication is defined entrywise, i.e. $(a,b)(a',b')=(aa',bb')$. This construction has nothing to do with pullbacks or monoids specifically, it holds for all limits of algebraic structures. See e.g. Borceux, Handbook of categorical algebra, or Adamek-Herrlich-Strecker, Abstract and concrete categories.