Is the category of monoids cartesian closed? Why?

Question

Is the category of monoids cartesian closed? Why?

I read Steve Awodey's "Category Theory", and could not solve the exercise in chapter 6, stated above.

Here I speak of the "category of monoids" as the category with objects monoids and arrows homomorphisms between monoids.

Answer 1

Whenever a non-trivial category is cartesian-closed, the final object $1$ cannot also be an initial object.

Otherwise:

$$\mathrm{Hom}(A,B)\cong \mathrm{Hom}(1\times A,B) \cong \mathrm{Hom}(1,B^A)\cong\{\cdot\}$$

That is, every hom set would have to be a singleton.

Answer 2

If $M$ is a non-trivial monoid, the functor $M \times - : \mathsf{Mon} \to \mathsf{Mon}$ doesn't preserve the initial object, and hence it cannot have a right adjoint.