The Problem is: Let, $G$ be a group, not necessarily abelian. Let $\mathcal{C}$ be the category of all based CW complexes. Show the functor $F \colon \mathcal{HoC}^{\mathrm{op}} \to \mathcal{Sets}$ given by $F(X) = \mathrm{Mor}_{\mathcal{Groups}(\pi_1(X),G)}$ is topologically exact. Also, show every exact functor from $\mathcal{HoC}^{\mathrm{op}}$ to $\mathcal{Sets}$ is of the form $F$.
My approach: Firstly, $F$ is topological half-exact functor by the Van-Kampen theorem. I think if $F' \colon \mathcal{HoC} \to \mathcal{Sets}$ is given by $F'(X) = F(X)$, then we can show that $F'$ is exact using Van-Kampen only as whenever $X = A \cup B$ with $A \cap B$ connected and containing the base-point, then $F'(X) \to S = F'(A) \times_{F'(A\cap B)} F'(B)$ so it becomes a pullback square of sets where the four maps are induced by various inclusions.
Also, any hints how to show that all exact functors are of that form? We can use Brown’s Representation Theorem here as all exact functors are half-exact and $F'$ is a functor from homotopy category of $\mathcal{C}.$
Will there be any application of Yoneda Lemma here?
Thanks in advance.