@Igor – I think that's a good rule-of-thumb but there are some gotchas to bear in mind.

• talking about \$$f\$$ being "injective" or "surjective" only makes sense in categories where the arrows are functions of some sort

• any function with a left inverse is injective, but not vice versa – consider the map \$$\varnothing \rightarrow \textrm{1}\$$

• any function with a right inverse is surjective, but the converse only holds if we have the Axiom of Choice