A quick question about lax co/limits.

Strictly, when $F : J\to \bf A$ is a diagram and $J$ has an initial object $\varnothing$, then $\varprojlim F \cong F(\varnothing)$; dually, if $\cal J$ has a terminal object, then $\varinjlim F\cong F(*)$.

If $F$ is a diagram between 2-categories (same notation), and $J$ has a lax initial, lax terminal object, is a similar statement true for lax co/limits?