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 $\varinjlim F\cong F(\*)$.



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> 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?
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Lax co/limit as evaluation on terminal/initial
No. If $\mathcal{J}$ is the interval category (two objects and one nonidentity morphism between them), which certainly has an initial and a terminal object, then the lax limits and colimits of such a diagram are comma objects of the morphism that determines its image, neither of which is usually equivalent to its domain or codomain.


