My book has this definition:
Let $ A $ be an object of a concrete category $ \scr{C} $, $ X $ a nonempty set, and $ i: X \to A $ a map (of sets). We say that $ A $ is free on a set $ X $ provided that for any object $ B $ of $ \scr{C} $ and a map (of sets) $ j: X \to B $, there exists a unique morphism $ \phi \in {\text{Hom}_{\scr{C}}}(A,B) $ such that $ \phi \circ i = j $ (composition of functions).
Question 1: Wikipedia's definition says that the map $ i $ is a canonical injection. The one above allows the map $ i $ to be not injective. I feel that a good definition must include the injectivity of $ i $; am I correct?
Question 2: Can the definition of free objects be extended to categories other than concrete categories? One definition I thought of is: Let $ \scr{C} $ and $ \scr{D} $ be two categories, and let $ F := (F_{0},F_{1}) $ be a functor from $ \scr{C} $ to $ \scr{D} $. Let $ A $ and $ X $ be objects in $ \scr{C} $ and $ \scr{D} $ respectively. (Notation: Here, $ F_{0} $ is the map between the objects of $ \scr{C} $ and $ \scr{D} $, and $ F_{1} $ is the map between the morphisms of $ \scr{C} $ and $ \scr{D} $). Let $ i \in {\text{Hom}_{\scr{D}}}(X,{F_{0}}(A)) $. We say that $ A $ is free on $ X $ iff for every object $ B $ in $ \scr{C} $ and morphism $ j \in {\text{Hom}_{\scr{D}}}(X,{F_{0}}(B)) $, there exists a unique morphism $ \phi \in {\text{Hom}_{\scr{C}}}(A,B) $ such that $ {F_{1}}(\phi) \circ i = j $. Again, I don't know what condition to add on the morphism $ i $ (if there is any to add).
As I am new to category theory, I don't know if it is interesting to create a definition for free objects in non-concrete categories (I just think it is).
Thank you all.