On the Size of Categories

The purpose is to give a simple proof that a category is equivalent to a small category if and only if both it and its presheaf category are locally small In one of his lectures University of New South Wales on Yoneda structures SW the second author conjectured that a category A is essentially small if and only if both A and the presheaf category PA are locally small The rst author was in the audience and at the end of the lecture suggested a proof of the conjecture using some of his own results This was reported on page of SW and used to motivate a de nition of small in St yet the proof was not published The proof given in the present paper evolved via correspondence between the authors in while the second author was on sabbatical leave at Wesleyan University Middletown Connecticut but has remained unpublished despite our expectation at various times that it would appear as an exercise in some textbook In a longer but related proof appeared in F We advised the author of this history and sent him our proof This was reported in F but our proof was still not published Now that there is actually an application RW we decided publication was in or der We have expressed the construction in a form we believe begs generalization to for example parametrized categories SS Note throughout that small can mean nite For an object A of a category A we let