# Definition of subobject

In the definition of subobject there is an equivalence relation defined on monomorphisms into a fixed codomain.

My question is that how do we know that the collection of monomorphisms into a fixed codomain is a set?

Is it obvious? or Is it because we can define equivalence relation not only on a set but a class? I don't know.

• We don't, it's not obvious, and we're defining the equivalence relation on a class. This issue crops up already in $\text{Set}$. . – Qiaochu Yuan Jun 3 '14 at 5:18
An example of a category which is not well-powered is the partial order $V \cup \{\infty\}$, where $V$ is some Grothendieck universe and $\infty$ is some new element with $x \leq \infty$ for all $x \in V$.