Is Pic a functor?

Question

Let $K$ be a field. Is $Pic$ a functor from the category of fields and field homomorphisms to the category of groups and group homomorphisms, where $Pic(K)$ is the Picard group of $K$?

Answer 1

The Picard group is even a (contravariant) functor on the category of schemes Sch. Since the category of commutative rings and ring homomorphisms CRng is a subcategory of $\mathbf{Sch}^\mathrm{op}$, and Fld is a subcategory of that, the yes, Pic is a (covariant) functor on Fld.

However, that is overkill; I mention it only to satisfy other related queries you might have.

As mentioned in the comments, the Picard group of a UFD is trivial. Since fields are UFDs, as a functor on Fld, Pic is isomorphic to the constant functor that assigns to every field $K$ the zero group, and to every field morphism the identity map on the zero group.