About argument of forgetful functor 
Let $(\{0,1\},0) \in \text{Set}_*$ be the set $\{0,1\}$, equipped with the base point $0 \in \{0,1\}$. Prove that $(\{0,1\},0)$ represents the forgetful functor $U:\text{Set}_* \rightarrow \text{Set}$.

Forgetful functor is used for any functor that forgets structure, and whose codomain is the category of sets. For example, $U: \text{Group} \rightarrow \text{Set}$ sends a group to its underlying set and a group homomorphism to its underlying function.
The only thing I can think of for argument is the codomain does not contain base point, so $(\{0,1\},0)$ represents the forgetful functor. How to improve on this? Thanks.

Given any pointed set $(S, *)$, what are the morphisms $(\{0, 1\}, 0)\to(S, *)$? They are precisely the functions $f\colon\{0, 1\}$ with $f(0)=*$ and with $f(1)\in S$ arbitrary. We may define a bijection $\varphi\colon\text{Hom}_*(\{0, 1\}, S) \to S$ (where the subscript signals that these are pointed functions) the following way: given any pointed function $f$, let $\varphi(f)=f(1)$. The inverse is given by $\theta(s)=f_s\colon \{0, 1\}\to S$, where $f_s(0)=*, f_s(1)=s$. I'll let you check the naturality as an exercise.
