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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.

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    $\begingroup$ We don't, it's not obvious, and we're defining the equivalence relation on a class. This issue crops up already in $\text{Set}$. . $\endgroup$ – Qiaochu Yuan Jun 3 '14 at 5:18
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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.

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