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.
Please help. Thank you.
It's not satisfied in general. This leads to the notion of a well-powered category.
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 $x \in V$.
See MO/93853 for abelian categories which are not well-powered.