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We have a category $\mathbb C$ with finite products and terminal object $1$. Further $\mathsf{Mon}(\mathbb C)$ is the category of monoids in $\mathbb C$ where a monoid is a triple $(M,m:M\times M \rightarrow M,e:1 \rightarrow M)$ which fulfills certian associativity and unit axioms. I have already proven that $\mathsf{Mon}(\mathbb C)$ has binary products. Assume $\mathbb C$ has equalizers. Now I have to prove that $\mathsf{Mon}(\mathbb C)$ has equalizers, too.

Further we have an arrow $f:A_1 \rightarrow A_2$ in $\mathsf{Mon}(\mathbb C)$ iff $f \circ m_1 = m_2 \circ (f \times f)$.

I would appreciate some good hints, how to start with this.

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Hint: Let's look at the example of monoids in set. Given $f,g \colon M_1 \to M_2$, the set where the maps coinside $\{m \in M \mid f(m) = g(m)\}$ [i. e. the equalizer in $\sf Set$] is a submonoid (as $f,g$ are morphisms). Together with the structure inherited from $M_1$ it is the equalizer in the category of monoids (here: semigroups with identity).

Let this example guide you in the general case, take the equalizer $E \to A_1$ in $\mathcal C$, restrict $m_1 \colon A_1^2 \to A_1$ to $E$ and show that the unit $e_1 \colon 1 \to A_1$ can be pulled back. (A monoid morphisms must also fulfill $f \colon e_1 = e_2$, right?)

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