The limit–terminality conjecture for pseudo- and lax-cones

Let $I$ and $\mathcal{A}$ be $2$-categories, and let $F\colon I\to\mathcal{A}$ be a $2$-functor. Let $L\in\mathcal{A}$ be an object and let $\lambda\colon\Delta L\Rightarrow F$ be a pseudo-natural (respectively, lax-natural) transformation.

Limit–terminality conjecture. The following two statements are equivalent:

$$\begin{array}{ll} \text{(i)} & (L,\lambda)\text{ is a pseudo-limit (respectively, lax-limit) of }F;\\ \text{(ii)} & (L,\lambda)\text{ is a }2\text{-terminal object in the strict-, pseudo-, or lax-slice }2\text{-category of pseudo-cones (respectively, lax-cones) over }F. \end{array}$$

The conjecture proposes that pseudo- and lax-limits can be characterized by $2$-terminality in the corresponding slice $2$-categories of cones. The surrounding discussion indicates that analogous claims for strict $2$-limits and lax- or pseudo-slices fail, and the paper subsequently investigates whether these weaker notions of limit yield the proposed relationships.