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!

Good question. No, it does not have a left adjoint. One of the most important properties of adjoints is that right adjoints commute with limits (including products and the terminal object), while left adjoints commute with colimits (including coproducts and the initial object). For example, since $\emptyset$ is an initial object and $X \times -$ is a left adjoint, $X \times \emptyset \cong \emptyset$.
On the other hand, if $X \times -$ were a right adjoint (that is, if it had a left adjoint), it would have to commute with limits, including the terminal object. In $\operatorname{FSet}$, the terminal object is $1$, a singleton set. However, in general $X \times 1 \ncong 1$. So $X \times -$ does not have a left adjoint (provided $X \ncong 1$).
