Let be a diagram in whose limit exists and such that . Then if is a cone in that is sent to a limiting cone in , then by the universal property of the limit in the morphism is an isomorphism in , hence must have been an isomorphism in , hence must have been a limiting cone in .
The arguments for colimits is analogous.
Geun Bin Im, George Maxwell Kelly, Some remarks on conservative functors with left adjoints, J. Korean Math. Soc. 23 (1986), no. 1, 19–33, MR87i:18002b, pdf; On classes of morphisms closed under limits, J. Korean Math. Soc. 23 (1986), no. 1, 1–18, Adjoint-triangle theorems for conservative functors, Bull. Austral. Math. Soc. 36 (1987), no. 1, 133–136, MR88k:18005, doi
For an example of a conservative, but not faithful, functor having a left adjoint see Example 2.4 in
Reinhard Börger, Walter Tholen, Strong regular and dense generators, Cahiers de Topologie et Géométrie Différentielle Catégoriques 32, no. 3 (1991), p. 257-276, MR1158111, numdam
Revised on April 12, 2011 16:45:14
by Zoran Škoda