So last post I gave a (hurried) description of why adjoint pairs of functors are linked to this notion of “similar structure” between two categories.  In this post, I want to relate adjunctions to universal properties, and ultimately why we like adjoint pairs so much.

Say we’re working with the “free group on a set” functor, $F: \textbf{Set} \to \textbf{Grp}$.  We know that the free group satisfies a really nice universal property: Given any (set) function $f: X \to G$, where $G$ is a group, there exists a unique group homomorphism $\overline{f} : F(X) \to G$ extending $f$.   If you recall the notion of the “forgetful” functor $U: \textbf{Grp} \to \textbf{Set}$ that takes a group and gives its underlying set, the universal property of the free group on $X$ states that there is a bijection $\text{Hom}_{\textbf{Grp}}(F(X),G) \cong \text{Hom}_{\textbf{Set}}(X, U(G))$

i.e. $F \dashv U$ is an adjoint pair (of course, one should check the naturality of this bijection).  Hence the universal property here is really just this adjunction in hiding.  This idea is generalized often by saying that a category C has “free objects” if a suitably defined forgetful functor from C possesses a left adjoint.

It’s for this reason that I like to think of a pair of adjoint functors $F \dashv G$ as a sort of “globally defined” universal property, as expressed by the naturality of the bijection between hom-sets.  Indeed, we have that a functor $G: \textbf{D} \to \textbf{C}$ has a left adjoint provided that we can find, for each object $A$ of an object “ $FA$” of D and a morphism $\eta_A : A \to G(FA")$ that is universal among morphisms form $A$ to the image of $G$, i.e. for all $f: A \to GB$, there is a unique $\overline{f} : FA" \to B$ satisfying $f = G(\overline{f}) \circ \eta_A$.  I use the quotations here only to emphasize the fact that we really want the object “ $FA$” to be “the component of some suitable functor $F$ at $A$.”

I don’t know about you, but this sound a lot like: “For any functor $S: \textbf{1} \to \textbf{C}$, the comma category $(S(*) \downarrow G)$ has an initial object.” (Remember, a functor $S: \textbf{1} \to \textbf{C}$ simply selects an element of C.)  That is, this sounds A LOT like how we initially (haha, pun) constructed universal properties from comma categories.  The statement of the above paragraph then says that if we can find such an initial object for any choice of functor $S: \textbf{1} \to \textbf{C}$, then $G$ has a left adjoint.  Pretty neat, right?