I am studying Category Theory on my own and find many examples and proofs use the fact that certain properties are invariant among all isomorphic objects within a given category. However, many such invariances are used without any prior exposition or proof - as if “everybody knows” which properties are isomorphic invariant.
Could anyone please enumerate the most useful invariant properties for isomorphic objects within a given category?
Since many objects such as posets and groups can also be viewed as self-contained categories, it would also be great to have a list of the most useful invariances of isomorphic categories.
Thanks