Name for categories in which isomorphic implies equal?

Question

A quick terminology question: Is there any particular name for a category in which each object is uniquely determined by its isomorphism class?

Answer 1

This article describes "skeletal" categories, which have the property you desire: http://en.wikipedia.org/wiki/Skeleton_%28category_theory%29

Answer 2

A "gaunt" category is similar, but does not allow any nontrivial automorphisms, which are allowed in a skeletal category. http://ncatlab.org/nlab/show/gaunt+category