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

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    $\begingroup$ 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. $\endgroup$ – Vladimir Sotirov Aug 12 '16 at 14:55
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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.

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