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!

In this category, I find counting to be effective at disproving conjectures.
If it did have a left adjoint $F$, then there would be a bijection
$$ \hom(FA, B) \cong \hom(A, X \times B) $$
so we would have
$$ b^{|FA|} = (|X| b)^{|A|} $$
for every finite cardinal $b$. Supposing $|X| > 0$, we would need
$$ |FA| = |A| (1 + \log_b |X|) $$
for all positive integers $b$. Clearly this cannot happen if $|X| > 1$.
