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I need some further help in order to fully understand some further aspects of Section 1.4 Example 13 ...

Section 1.4 Example 13 reads as follows:View attachment 8347

View attachment 8348

In the above text by Awodey we read the following:

" ... ...In detail, a homomorphism from a monoid \(\displaystyle M\) to a monoid \(\displaystyle N\) is a function \(\displaystyle h : M \to N\) such that for all \(\displaystyle m,n \in M\),\(\displaystyle h( m \bullet_M n ) = h(m) \bullet_N h(N) \)

and

\(\displaystyle h( u_M ) = u_N \)

Observe that a monoid homomorphism from \(\displaystyle M\) to \(\displaystyle N\) is the same thing as a functor from \(\displaystyle M\) regarded as a category to \(\displaystyle N\) regarded as a category. ... ... "I cannot see how (exactly and rigorously) the monoid homomorphism specified above fits the 3 conditions (a), (b) and (c) that Awodey lays down for a functor (see text below) ... ...

Can someone please demonstrate explicitly and rigorously how (exactly) the monoid homomorphism specified above fits the 3 conditions (a), (b) and (c) that Awodey lays down for a functor (see text below) ... ...Help will be appreciated ... ...

Peter=======================================================================================

*** NOTE ***

In order to help and answer the question posed in the above post, MHB readers of the post need access to Awodey's definition of a functor ... so I am providing access to the same ... as follows:View attachment 8345

https://www.physicsforums.com/attachments/8346Hope that helps ...

Peter