TITLE: Fundamental Group as a Functor and its Adjoint
QUESTION [9 upvotes]: So taking the fundamental group of a pointed topological space can be thought of as applying a functor, $\pi_1$, that gives you something in the category of groups.  Does $\pi_1$ have an adjoint?

REPLY [6 votes]: If you see $\pi_1$ as a functor from the homotopy category of pointed connected CW-complexes to the category of groups, it is a left adjoint (that does not contradict Ronnie Brown's answer). See https://mathoverflow.net/q/109779/24563 or https://mathoverflow.net/a/45361/24563. The right adjoint is the classifying space functor.