Anyone aware of a nice counterexample to "The fundamental group functor is full?" (Which is...false, right?) and are there a nontrivial subcategories on which its restriction is full?
I.e. Can you think of an example of topological spaces $X$ and $Y$ such that there is a group homomorphism $\phi: \pi_1(X, x_0) \to \pi_1 (Y,y_0)$ such that $\phi$ is not induced by any continuous map $f: X \to Y$ ?
Sometimes every homomorphism is induced by a continuous map: For example every automorphism of $\pi_1(S^1)$ is induced by a map $S^1 \to S^1$. Are all automorphisms of fundamental groups induced by continuous maps? What conditions on spaces $X$ and $Y$ ensure every fundamental group homomorphism is induced by a continuous map?