# Word for category with unique isomorphisms

Is there a standard word to express the fact that a category has at most one isomorphism between any two objects?

• You could say that the groupoid of isomorphisms in the category is simply-connected. This means that every connected component of the groupoid is $1$-connected, which says that there is exactly one isomorphism between any two objects in the connected component. – Vladimir Sotirov Aug 12 '16 at 14:55

This is equivalent to being gaunt, a common name for a category with no nontrivial automorphisms. A non-gaunt category certainly has a pair of objects admitting two isomorphism a between them, while if $f,g$ are distinct isomorphisms in any category then $g^{-1}f$ is a non-identity automorphisms, by uniqueness of inverses.