Finn Lawler 2-end

I think the easiest way to define a bicategorical notion of end is to follow the identification in enriched category theory of ends with $\hom$-weighted limits.

The Yoneda lemma for 2-extranatural transformations shows that for a biprofunctor $H \colon K^{op} \times K \to Cat$ there is an equivalence $Exnat(\mathbf{1}, H) \simeq \{\hom_K, H\}$. For an arbitrary $T \colon K^{op} \times K \to L$, we then find that $Exnat(\ell, T) \simeq L(\ell, \{\hom_K, T\})$ if that limit exists, so that we may write this limit as $\int_a T(a,a)$.

One fact will be useful, and it is easy to show: for pseudofunctors $F, G \colon K \to L$, the category $Nat(F,G) = [K,L](F,G)$ is equivalent to $\int_a L(F a, G a)$ as usual in enriched category theory. This is easiest to see by comparing $Nat(F,G)$ with $Cat(\mathbf{1}, \int_a L(F a, G a)) \simeq Exnat(\mathbf{1}, L(F-, G-))$.