If $F: \mathscr{A}\to \mathscr{B}$ is a functor and $[\mathscr{C}, \mathscr{D}]$ is the category of functors and natural transformations between two given categories $\mathscr{C}$ and $\mathscr{D}$ we have the usual composition functors $F^*: [\mathscr{B}, \mathscr{C}] \to [\mathscr{A}, \mathscr{C}]$, $F_*: [\mathscr{C}, \mathscr{A}] \to [\mathscr{C}, \mathscr{B}]$.
Consider $F^*: [\mathscr{B}, \mathscr{C}] \to [\mathscr{A}, \mathscr{C}]$, if $\mathscr{C}$ is complete (i.e. has small limits) then also $[\mathscr{B}, \mathscr{C}] $ and $[\mathscr{A}, \mathscr{C}]$ are complete and the limits are pointwise and then $F^*$ is complete (i.e. preserve small limits). There exists example of no pointwise limits and functors $F$ such that $F_*$ isn't complete (see "Basic Concepts of Enriched Category Theory" G. Kelly, p.77 in the old edition).
Then I consider the 2-category $CAt_c$ of complete categories (by functors and natural transformations)
Then (with the above restriction) functors like $F^*$ are always complete, but of course $F_*$ isn't always complete (consider $\mathscr{C}=1$ and a no limit preserving $F$).
Question.2: How can explain (in 2-categorical terms and property, I thought about Kan extension..) this asymmetry about composition of morphism in the 2-category $CAt_c$?
In formal terms I looking for a property $\mathcal{P}$ about 2-categories such that it is true for $CAt$, and given a 2-category $\mathscr{C}$ with the property $\mathcal{P}$, then for $f: A\to B$ and $C$ in $\mathscr{C}$, the composition functor $f^*: [B, C]\to [A, C]$ is complete.
