![transpose identity](http://aether.co.kr/images/transpose_identity.svg)

Okay I think I kind of get it now. So a square matrix is an identity morphism in \$$\mathbf{FinVect}\$$. \$$AA^\mathsf{T}\$$ is always a square matrix so will always be equal to an identity morphism. I kept thinking a identity matrix was the identity morphism but in this category, the collection of square matrices is the identity morphisms which threw me off.