After returning home and checking my references, I see that in [Blackwell, Kelly, Power] we show that, for a 2-monad T on a complete 2-category K, the category T-Alg, whose morphisms are the non-strict ones (the "to within a coherent isomorphism" ones), admits products, inserters, and equifiers, formed as in K, and hence such limits as follow from these, including the cotensor product (-)^X for an X in K. What I said today about admitting all FLEXIBLE limits is false; I had misremembered, and there is a counter- -example in Example 6.2 of [Bird, Kelly, Power, Street, Flexible limits for 2-categories, JPAA 61 (1989), 1 - 27]. The flexibles would follow from those above if idempotents were to split - but they don't in general. I am strongly convinced that the above is true. The mere CATEGORY Ccc of cartesian closed categories was shown to be monadic over the mere category Cato of categories by Burroni, and this was checked and restated by Dubuc and Kelly in [J. Algebra 81 (1983), 420 - 433]. Blackwell, Kelly, and Power claimed on p.39 of [B,K,P] that Ccc, as a 2-category with only invertible 2-cells, is monadic (2-monadic, if you want the repeated emphasis) over the complete 2-category Catg which is obtained from the 2-category Cat by discarding all the non-invertible 2-cells. It is most unlikely that Ccc is 2-monadic over the usual 2-category Cat; for already symmetric monoidal closed categories is known not to be so; see p.34 of [B,K,P]. I had strongly believed that Ccc was indeed 2-monadic over Catg; but we never checked every detail of this particular example. Today my belief has been shaken by two letters (one from Barry Jay) claiming that the monadicity of Ccc is false, unless the cartesian closed categories are supposed to admit pullbacks. I shall try to check this out in the coming days - we were quite sure that the structure is equational even without pullbacks, but perhaps we were wrong. Certainly Burroni (see [Dubuc-Kelly for the reference) supposed pullbacks present, because he was headed for toposes. I must say that the letters I received today were imprecise about which base 2-category was involved; to be fair, they were not really about monadicity but about the existence of A^2. However I formed theimpression that the 2-categorical investigations in the references above have not (yet?) become part of the common language of our colleagues. Note that the T-Alg where these limits are to exist does NOT have the strict maps; so that it is certainly not monadic in any classical sense. Clearly one cannot begin even to pose such questions precisely without using 2-categorical notions. Let me finish by saying that the universal property of A^2 is 2-categorical by its very nature (if we are speaking of A^2 in T-alg, not just in Cat). Oh, that's it! I have just seen what the point is. [B,K,P] is quite right in asserting that Ccc has the limit A^2 formed as in Catg; but the limit A^2 in Catg is not at all that in Cat; it is the category such that to give B --> A^2 is to give two maps B --> A and a 2-cell between them; but the 2-cells in Catg are just the INVERTIBLE natural transformations, so that A^2 in Catg is A^I in Cat, where I is the free isomorphism, not the free a arrow 2. So, all right, I give in. The [B,K,P] results don't give a cartesian closed structure on the USUAL A^2 in Cat. Anyway, I've learned (or relearned) something from thinking this through, and hope some others may benefit from seeing these notions brought up. And so to bed. Max Kelly.