Definition of adjoint functor and locally small categories In the definition of an adjoint pair of functors, is it implicit that the categories are locally small? I have searched for ages, and nowhere is this stated as an assumption, but the definition seems to require it. (Where we take the definition in terms of Hom isomorphism, for example here: http://ncatlab.org/nlab/show/adjoint+functor )
 A: There is another way of stating an adjunction without referring to hom-sets, due to Lawvere (I think).
A functor $F \colon \mathcal A \to \mathcal B$ is left adjoint to $G \colon \mathcal B \to \mathcal A$ if and only if there exist an isomorphism of categories $\left(F \downarrow \mathrm{id}_{\mathcal B} \right) \stackrel \phi \simeq \left( \mathrm{id}_{\mathcal A} \downarrow G \right)$ between the comma categories making the following diagram commute:
$$
\begin{matrix} 
  \left(F \downarrow \mathrm{id}_{\mathcal B} \right) 
  & \stackrel \phi {\large\simeq} 
  & \left( \mathrm{id}_{\mathcal A} \downarrow G \right) \\
  \hskip 20pt \searrow & & \hskip -20pt \swarrow \\
  & \mathcal A \times \mathcal B & 
\end{matrix}
$$
This definition makes sense even if the categories are not locally small.
A: The unit/counit definition of an adjunction works for categories which are not locally small.
In fact, the notion of an adjunction may be defined in an arbitrary $2$-category, see here.
We don't need Hom-sets. On the other hand, as stated in the comments, you may also work with Hom-classes to describe adjunctions between categories which are not locally small.
Notice. In many books categories are assumed to be locally small by definition. There are various reasons for this, but I guess this is not the best place to explain all of them.
