The $2$-cobordism category $2\text{Cob}$ is the category whose objects are (compact, oriented) $1$-manifolds and whose morphisms $M \to N$ are (compact, oriented) $2$-manifolds with boundary $M \sqcup N$ in the appropriate orientation up to relative homeomorphism, with composition given by identifying the appropriate parts of the boundary.
$2\text{Cob}$ is symmetric monoidal with the monoidal operation given by disjoint union. But in fact much more is true.
Theorem: $2\text{Cob}$ is the free symmetric monoidal category on a commutative Frobenius object.
In other words, symmetric (strong?) monoidal functors $2\text{Cob} \to M$, where $M$ is a symmetric monoidal category are the same as commutative Frobenius algebras in $M$.
I always thought this one was particularly funny because the original motivation for the definition of a Frobenius algebra has nothing to do with topology. This is a very special case of the cobordism hypothesis in TQFT and is explained carefully with lots of pictures in Kock's Frobenius algebras and 2D topological quantum field theories.
