# Are bicategories of lax functors also bicategories of of pseudofunctors?

Let A be a bicategory. Can I construct in some "natural" way a bicategory L(A) such that for every bicategory B, the bicategory of lax functors Lax(A, B) is (bi)equivalent to the bicategory of pseudofunctors Pseudo(L(A), B)? (Choose the morphisms in these functor categories as you wish. Ideally the same construction L would work for any choice.)

For example, when A = •, I believe we may take L(A) = BΔ+, the delooping of the augmented simplex category with monoidal structure given by ordinal sum. In general I imagine L(A) as being formed as something like the free category on the objects, 1-morphisms and 2-morphisms of A--denote the generator corresponding to a 1-morphism f of A by [f]--as well as 2-morphisms id → [id] and [f] o [g] → [f o g] for every composable f and g in A.

(By Chris's question here, L(A) cannot be an equivalence invariant of A.)

• Reid, I think there's a typo. You should have "Pseudo(L(A), B)" in the second sentence. – Tom Leinster Dec 4 '09 at 12:44