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I know that left adjoints preserve colimits and right adjoints preserve limits. So clearly if the limits (resp. colimits) in both categories exist, the adjoints map them to each other.

My question is, whether it's garantueed that the limit exists?

To be more precise: Let $T\colon A\rightarrow B$ a functor that is a right (resp. left) adjoint of some functor and $D\colon I\rightarrow A$ a diagramn, such that it's limit (resp. colimit) (in $A$) exists. Does the limit (resp. colimt) $T\circ D$ always exist?

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Yes, and this is precisely the statement that $T$ preserves limits! If $\{L \to D\}$ is a limit cone, and $T$ preserves limits, then $\{T(L) \to T \circ D\}$ is a limit cone.

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  • $\begingroup$ Thank you. I was only aware of the "preserveness" if both limits do exist. $\endgroup$ – John May 25 '13 at 12:57

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