A question about the product functor on finite sets I am a beginner in Category Theory so please excuse me if this is a trivial question.
Let $\mathbf{FSet}$ denote the category of finite sets. The product functor $X\times -:\mathbf{FSet}\to \mathbf{FSet}$ has a right adjoint for every finite set $X$. 
My question is, does it also have a left adjoint?
Thanks!

No, it doesn't unless $X$ is a singleton. The very first condition to check for a functor to have a left adjoint is that it should preserve limits (such as products, equalizers...). But clearly in general, if $X$ has at least two element,
$$X \times (Y \times Z) \not\cong (X \times Y) \times (X \times Z),$$
and so the functor doesn't preserve products, thus it doesn't have a left adjoint.
If $X = \varnothing$ we do have the equality above, but then $\varnothing \times \{*\} = \varnothing \neq \{*\}$ and the functor does not preserve the terminal object. So again it doesn't have a left adjoint.
However if $X = \{*\}$ itself then your functor is the identity functor, which indeed has a left adjoint: itself.
