I'm studying this book on introductory level category theory and I couldn't solve this exercise:
In the first part I've been thinking about the monoid homomorphisms $F: S\to T$ and regarding of second part I don't know even how to begin.
I need help
thanks in advance
A covariant functor $F$ from $\tilde S$ ($S$ viewed as a category) to $\tilde T$ obviously sends the unique object of $\tilde S$ to the unique object of $\tilde T$. It maps "morphisms" in $\tilde S$ (that is, elements of the monoid $S$) to "morphisms" in $\tilde T$, and therefore gives a function $f : S \to T$. Finally, the functor satisfies $F(a \circ b) = F(a) \circ F(b)$ for every pair of morphisms in $\tilde S$; considering that composition is multiplication in $S$ (resp. $T$), it means that $f(ab) = f(a)f(b)$. Finally, a covariant functor here is exactly the same thing as a homomorphism of monoids.
For a contravariant functor, things are identical, except that in the end, $f(ab) = f(b) f(a)$. Such a thing is called a antihomomorphism (of monoids).