A question about covariant functor Let $X$ be a fixed set and define a functor $f\colon S \to S$ by $Y\mapsto X \times Y$. Then is $f$ a left adjoint of the covariant $\operatorname{Hom}$ functor $h_X=\operatorname{Hom}_s(X,-)$?
First, I don't think $h_X$ is a covariant functor from $S$ to $S$. Second,is $X\times Y$ an object in $S$? I feel confused.
 A: I assume that by $S$ you mean the category of sets and functions of sets.
Now first, if you have a function $\phi\colon A \to B$ then for $h_X$ to be covariant this means $\phi$ should induce a function $\hom(X, A) \to \hom(X, B)$.  And indeed it does.  This function maps $X \overset{\psi}{\to} A$ to the composition $X \overset{\psi}{\to} A \overset{\phi}{\to} B$.  I recommend that you check that the appropriate axioms are satisfied.
Second, if $X$ and $Y$ are sets then the cartesian product $X \times Y$ is certainly a set, so yes $X \times Y$ is an object in $S$.
Now for $(f, h_X)$ to be an adjoint pair you need an isomorphism
$$\hom(X \times A, B) \simeq \hom(A, \hom(X, B))$$
which is functorial in $A$ and $B$.  So you need to take a function $\phi\colon X \times A \to B$ and construct a function $A \to \hom(X, B)$.  How about sending $a \in A$ to the function $\phi(-, a)\colon X \to B$?  I'll leave you to check that this gives an isomorphism between those hom sets and is appropriately functorial in $A$ and $B$.
