Is there a category theory condition such that we can say when a property of a category also holds for it's sub-category?
For example, in the catgeory of groups $f$ being bijective implies it is an isomorphism, and in the sub-categories of rings, modules, algebras and so forth. However this is not the case for example for topological vector spaces.
My question is there a categorical condition (aside from simply defining a new term), that can ensure a property will hold for a sub-category?
Yes, one example is that a full subcategory of an abelian category is again abelian (under certain additional conditions):
Full subcategory of abelian category is abelian