In the paper I am now writing most of the categories are locally small (so, object form a class and $Mor(X,Y)$ is always a set). Yet they are not small, and sometimes I have to mention something like the category of additive functors from $C$ into abelian group. What can I say about the "size" of this functor category (to note that it is not locally small in contrast to other categories used in the paper)? What can one say about the functor category if $C$ is essentially small but not small?

I appears that to treat categories of functors "properly" one has to use Grothendieck universes. Yet the papers (and a book) on triangulated categories that I cite in my text (almost) ignore this matter, and I don't want to pay much attention to it either. So I wonder which terminology can I use and which "precautions" should be taken so that my simple arguments concerning functor categories will be mathematically correct. In particular, if the restriction of certain functors to a small subcategory of $C$ gives an equivalence of functor categories then can I say that the original category of functors is locally small?

In the paper I am now writing most of the categories are locally small (so, object form a class and $Mor(X,Y)$ is always a set). Yet they are not small, and sometimes I have to mention something like the category of additive functors from $C$ into abelian group. What can I say about the "size" of this functor category (to note that it is not locally small in contrast to other categories used in the paper)? What can one say about the functor category if $C$ is essentially small but not small?

I appears that to treat categories of functors "properly" one has to use Grothendieck universes. Yet the papers (and a book) on triangulated categories that I cite in my text (almost) ignore this matter, and I don't want to pay much attention to it either. So I wonder which terminology can I use and which "precautions" should be taken so that my simple arguments concerning functor categories will be mathematically correct. In particular, if the restriction of certain functors to a small subcategory of $C$ gives an equivalence of functor categories then can I say that the original category of functors is locally small?

Do you know any text that introduces "the most conventional" terminology for these matters? I would not like to mention universes, and I don't like terms like "small set".