A category being locally small does not imply its subcategory is locally small

Question

Mitchell's Theory of Categories defines locally small category as the following. Let $C$ be a category and $A\in Obj(C)$. Denote $S=\{[B]\subset A\}$ as the class of equivalent subobjects of $A$. If $S$ is a set, then one calls $C$ a locally small category. (It seems that this is a stronger requirement than the wikipedia's locally small category which only requires Hom sets as sets.)

The point is to look for a category $C$ which is locally small but its subcategory $C'$ is not locally small.

There are 2 cases I want to distinguish to allow $C'$ not locally small.

1. There are more monomorphisms in $C'$ than $C$

2. The two mono maybe isomorphic in $C$ but not in $C'$.

$\textbf{Q:}$ What are corresponding example of $C,C'$ in case 1 and case 2 separately? It seems I need to somehow run into russel paradox type construction but I do not see the obvious examples. However, if $C$ is a set category, there is no way I can get 1 to work. My guess for 2 would be considering $C$ to be set category and to get $C'$ not locally small via 2, I need to delete enough invertible elements to generate more than enough inequivalence classes but I am not sure cardinality will work out in general.

Answer 1

First some terminology. Wikipedia's definition of locally small is the usual one (so every $\operatorname{Hom}(A, B)$ is a set), and what you are talking about is usually referred to as well-powered. This is also the terminology I will use in this answer. Neither of those imply the other. In fact, one of the examples below is locally small, but not well-powered (example 2). It should also be clear that in this sense every subcategory of a locally small category is locally small.

So your question is then: can we make a well-powered category into a non well-powered category by deleting arrows such that either

1. some arrows become mono while they were not before,
2. some monos no longer induce the same subobject.

We can do the following for each case.

1. Consider a category with as objects $X$, $Y$, and a proper class of objects $A_i$. For each $A_i$ we have two different arrows $X \to A_i$ and just one arrow $A_i \to Y$. We have one arrow $X \to Y$ and let the composition of any $X \to A_i \to Y$ be that arrow. Of course, we also have identity arrows. Then the only monos are the identity arrows, $X \to Y$ and every $X \to A_i$. If we now delete all the arrows $X \to A_i$ we suddenly have that each $A_i \to Y$ becomes mono, and will represent its own subobject. So we get a proper class of subobjects.
2. Start with the category of sets. Delete all arrows, except for identity arrows and arrows that have a singleton as domain and not a singleton as codomain. Fix some set $X$ with at least two elements. Then any two singletons will represent different subobjects of $X$. There is a proper class of singletons (e.g. for every set $Y$ there is a singleton $\{Y\}$), so $X$ has a proper class of subobjects.

You can also combine these two approaches to get examples. Might be nice to think about that yourself.