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


*

*There are more monomorphisms in $C'$ than $C$

*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. 
 A: 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


*

*some arrows become mono while they were not before,

*some monos no longer induce the same subobject.


We can do the following for each case.


*

*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. 

*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.
