$\textbf{C}$-Monoids and products

Question

i have a question about $\textbf{C}$-Monoids. We can make a new category $\textbf{Mon(C)}$ from the category $\textbf{C}$, namely the category of all $\textbf{C}$-monoids. A $\textbf{C}$-monoid is a triple $(A,m:A\times A\rightarrow A, e:1\rightarrow A)$ with $1$ a terminal object and $m$ associative and $e$ a unit. The statement i want to prove is that $\textbf{Mon(C)}$ has binary products. But how to define the multiplication in a product of two $\textbf{C}$-monoids? And how to show the uniqueness etc. The product and the UMP are clear for me (thus the definition) but this is a big step. Can someone help me?

Answer 1

Let $(A,m,e)$ and $(A',m',e')$ be $\bf C$-monoids. Then their product will be $$(A\times A',\,\bar m,\, (e,e'))$$ where $\bar m$ is composed with the canonical isomorphism $$A\times A'\times A\times A' \ \overset{\cong}\longrightarrow\ A\times A\times A'\times A'\ \overset{(m,m')}\longrightarrow\ A\times A'\,.$$