If you are doing serious category theory, then at some point you will come across what are affectionately known as 'size considerations' or similar. In particular, any presheaf category $Cat(C,Set)$ and the subcategory of sheaves is not locally small (homs are sets) when $C$ is not a small category (set of objects). For example, you might want to consider the category of sheaves on the category of spaces, or schemes, or on a topos (these are not usually small). Then the Yoneda embedding, as Ryan points out, will not work, which is a bit of a problem.

One workaround is the axiom of universes, say with two universes $U \in V$. Then you can talk about locally small categories in $U$ - homs are elements of $U$ whereas the objects form a subset of $U$ (so these categories are '$U$-large'). Then the presheaf category consists of functors to the category of sets which are (isomorphic to) elements of $U$. The (pre)sheaf category is then locally small in $V$, and the Yoneda embedding for this category is taken into presheaves with values in the category of sets which are (isomorphic to) elements of $V$.

Whenever you see the phrase 'locally small', you can be sure someone is using some sort of foundations that distinguishes between large and small - Universes, GBN class-set theory or similar - to get around the issue.

Edit: Actually if one wants to think of schemes as sheaves on CRing, then you need to think hard about local smallness, otherwise the category of schemes will not be category under the naive definition that the homs are sets.

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If you are doing serious category theory, then at some point you will come across what are affectionately known as 'size considerations' or similar. In particular, any presheaf category $Cat(C,Set)$ and the subcategory of sheaves is not locally small (homs are sets) when $C$ is not a small category (set of objects). For example, you might want to consider the category of sheaves on the category of spaces, or schemes, or on a topos (these are not usually small). Then the Yoneda embedding, as Ryan points out, will not work, which is a bit of a problem.

One workaround is the axiom of universes, say with two universes $U \in V$. Then you can talk about locally small categories in $U$ - homs are elements of $U$ whereas the objects form a subset of $U$ (so these categories are '$U$-large'). Then the presheaf category consists of functors to the category of sets which are (isomorphic to) elements of $U$. The (pre)sheaf category is then locally small in $V$, and the Yoneda embedding for this category is taken into presheaves with values in the category of sets which are (isomorphic to) elements of $V$.

Whenever you see the phrase 'locally small', you can be sure someone is using some sort of foundations that distinguishes between large and small - Universes, GBN class-set theory or similar - to get around the issue.

1

If you are doing serious category theory, then at some point you will come across what are affectionately known as 'size considerations' or similar. In particular, any presheaf category $Cat(C,Set)$ and the subcategory of sheaves is not locally small (homs are sets) when $C$ is not a small category (set of objects). For example, you might want to consider the category of sheaves on the category of spaces, or schemes, or on a topos (these are not usually small). Then the Yoneda embedding, as Ryan points out, will not work, which is a bit of a problem.

One workaround is the axiom of universes, say with two universes $U \in V$. Then you can talk about locally small categories in $U$ - homs are elements of $U$ whereas the objects form a subset of $U$ (so are '$U$-large'). Then the presheaf category consists of functors to the category of sets which are (isomorphic to) elements of $U$. The (pre)sheaf category is then locally small in $V$, and the Yoneda embedding for this category is taken into presheaves with values in the category of sets which are (isomorphic to) elements of $V$.

Whenever you see the phrase 'locally small', you can be sure someone is using some sort of foundations that distinguishes between large and small - Universes, GBN class-set theory or similar - to get around the issue.