Subobject

Remarks

Since $m'$ is a monomorphism, the factor $k$ in $m = m' \circ k$ is unique when it exists. If $m \equiv m'$, the two factors are inverse isomorphisms, so $m \equiv m'$ holds exactly when $m = m' \circ k$ for an isomorphism $k$. The relation $\le$ induces a partial order on the subobjects of $X$.

Example: in the category of sets, the monomorphisms are the injective functions. An injective $m \colon A \to X$ is equivalent to the inclusion of its image $m(A) \subseteq X$, since $m$ is a bijection onto $m(A)$. So the subobjects of $X$ correspond to the subsets of $X$.

Non-example: the map $\{a, b\} \to \{1\}$ that sends both points to $1$ does not represent a subobject of $\{1\}$, since it is not injective and hence not a monomorphism of sets.