There is a useful simple result related to this question:

Suppose $X: C \to D$ is a fibred category, such that $X$ has a right adjoint, and $C$ is cocomplete. Then for each object $J$ of $D$ the inclusion $C_J \to C$ of the fibre over $J$ into $C$ preserves colimits of connected diagrams. See the proof of Theorem B.1.7 on p. 579 of *Nonabelian algebraic topology* EMS Tracts in Mathematics Vol 15 (pdf downloadable from my web page). Actually the conclusion is true without the assumptions, but this useful case has a short proof, given there.

This usefully applies to pushout diagrams.