As is well known, the definition of an algebra can be generalised to the notion of an algebra $A$ in an monoidal category $C$ with finite sums. What I would like to know is can the notion of generating subset of an algebra be generalised to this context?

Moreover, is this definition invariant under equivalence of categories, i.e. if $A$ is generated by a subject object $S$ in $C$, which is equivalent to $D$ by a functor $F$, then is $F(A)$ generated by $F(S)$?

Finally, what is a good reference for all this?