Notes on Lax Ends

In enriched category theory, the notion of extranatural transformations is more fundamental than that of ordinary natural transformations, and the ends, the universal extranatural transformations, play a critical role. On the other hand, 2-category theory makes use of several other natural transformations, such as lax and pseudo transformations. For these weak transformations, it is known that we can define the corresponding extranatural transformations or ends. However, there is little literature describing such results in detail. We provide a detailed calculation of the lax end, including its relation to the lax limits. We prove the bicategorical coYoneda lemma as the dual of the bicategorical Yoneda lemma, and also show that the weight of any lax end is a PIE weight, but it might not be a weight for a lax limit.