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

Question

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