This question is a sort of continuation of a previous one.
In CWM, Maclane says
... every quotient object of a group $G$ in $\mathsf{Grp}$ is represented by the projection $\pi:G\rightarrow G/N$ of $G$ onto the factor group $G/N$ of $G$ by some normal subgroup $N$ of $G$, and $G/M\leq G/N$ holds if and only if $M \supset N$...
It's known that equivalent quotient objects have isomorphic codomains (in the original category). Here in $\mathsf{Grp}$, Maclane says that $|M|=|N|\iff G/M\equiv G/N$ (here $G/M, G/N$ stand for quotient objects). But according to the statement above, $G/M\equiv G/N$ implies their codomains - the actual factor groups $G/M,G/N$ - are isomorphic as groups. We seem to have obtained the result $|M|=|N|\implies G/M\cong G/N$ for normal $M,N\subset G$, which is false, as shown for instance here. What am I missing here?
Also, it seems normal subgroups of a fixed group $G$ are not in bijection with its quotient objects. Is this true, and if it is, what's "obstructing" the bijection we would naively want?
The question is answered by Zhen Lin in the comment. Morphisms of subobjects (resp. quotients) are defined as morphisms of the underlying objects which are compatible with the inclusion morphism (resp. projection morphism). And of course Mac Lane mentions this.