# Do proofs using the Yoneda lemma work for Categories that are not locally small?

I often see proofs in category theory using the Yoneda lemma. I wonder if these apply to general categories(categories in which the hom-sets need not be sets) or there is a tacit assumption that the category is locally small. Is there some generalisation of the Yoneda lemma to General categories that might make these proofs work? Examples: https://mathoverflow.net/q/80797 Right adjoints preserve limits

• I don't know whether that's always true, but most of the time, categories are implicitly assumed to be locally small Feb 13 '18 at 21:20
• The problem with not locally small categories is that the $\hom$ functors -which the Yoneda lemma is about - might lead out of $\bf S{et}$. So, we would need to define a 'big' category of classes, whatsoever, or choose to see the whole picture from a bigger Grothendieck universe, and then the original Yoneda is already working.. Feb 13 '18 at 22:37
• So, can I safely assume all categories I deal with to be locally small? Feb 14 '18 at 4:36
• @Jehu314 Usually "category" is explicitly defined to correspond to a locally small category, but it is quite easy to produce "categories" that are not locally small. For example, the functor "category" from $\mathbf{Set}$ to $\mathbf{Set}$ is not locally small. This is part of the motivation for Grothendieck universes. With them, you will always have more "room" for such constructions, and you can always think of them in normal set theoretic terms, just with respect to different elements of the Grothendieck universe. Feb 14 '18 at 14:50
• Are all these results true for general categories? Feb 15 '18 at 15:39