As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of generating subset of a monoid can be generalised to this context - precisely, by generating subset I mean a subset $S$ such that the smallest ideal of $A$ containing $S$ is $A$.

I would naively guess that one would need the existence of infinite sums in $C$, define a generating set to be a sub-object $X$ of $A$, such that there is an isomorphism $$ A \simeq \bigoplus_{i \in I} B_i \otimes S \otimes B'_i, $$ for some objects $B_i,B'_i \in C$.

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?