Given a bicategory $K$, Niles Johnson and Donald Yau define an initial component lax terminal object in $K$ to be an object $T$ together with a lax transformation $k: 1_K\to \Delta_T$ such that $k_T$ is the identity of $T$ and each $k_X$ is initial in the category $K(X,T)$.
I have no idea why they do this, and where the name comes from. Is there some way to motivate the definition, or to relate it to the definition of biterminal objects in a bicategory? Why does the initial component lax terminal object have the name it has? Especially what is the terminal part of the name about?
I believe that every biterminal object $T$ comes with pseudonatural transformation $1_K \to \Delta_T$, but I don't know much more. The reference is chapter 7 of the book 2-dimensional categories by said authors.