The following is from Problem 13 of Chapter 3 in Awodey's Category Theory.
13-Show that the category of monoids has all coequalizers as follows.
- Given any pair of monoid homomorphisms $f, g: M \rightarrow N$, show that the following equivalence relations on $N$ agree:
(a) $n \sim n^{\prime} \Leftrightarrow$ for all monoids $X$ and homomorphisms $h: N \rightarrow X$, one has $h f=h g$ implies $h n=h n^{\prime}$,
(b) the intersection of all equivalence relations $\sim$ on $N$ satisfying $f m \sim g m$ for all $m \in M$ as well as $$ n \sim n^{\prime} \text { and } m \sim m^{\prime} \Rightarrow n \cdot m \sim n^{\prime} \cdot m^{\prime} $$ 2. Taking $\sim$ to be the equivalence relation defined in (1), show that the quotient set $N / \sim$ is a monoid under $[n] \cdot[m]=[n \cdot m]$, and the projection $N \rightarrow N / \sim$ is the coequalizer of $f$ and $g$.
i can solve "1". how we can solve "2" ?