Right adjoint of covariant hom functor

Question

I've constructed the left adjoint of the functor $\mathbf{Hom(A, -)}: \mathbf{Sets} \to \mathbf{Sets}$. Now I'm trying to prove that the functor does not have a right adjoint, but I'm not sure how to do this.

I have a feeling I should the fact that right adjoints preserve limits and left adjoints preserve colimits.

Answer 1

If I remember correctly, right adjoints preserve limits, but if the occasion arises where $\bf{A}$ is not terminal then $\text{Hom}(-,\bf{A})$ doesn't perserve since $\mathrm{Hom}(\{*\},A)\neq\{*\}$.

If $\bf{A}$ is terminal then the functor is constantly equal to $\{*\}$ and then has a left adjoint that is constantly equal to $\varnothing$.

Exactly the same reasoning can be applied to show that $ - \bigoplus \bf{A}$ does not have adjoints unlesss $\bf{A}=\varnothing$.

Answer 2

You are on the right track. If this functor had a right adjoint, it would be a left adjoint and would commute with colimits.

Hint. Consider that $2 = 1 \sqcup 1$ and check the images of your functor on 2 and 1. ($1$ is the singleton set and $2$ is a set with two distinct elements.)