When is the category of representations compact?

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I already know that given a compact Lie group $G$ it's category of (continuous unitary) representations Rep$(G)$ is compact, i.e. allows duals. However I can't seem to find a good reference about the case of non-compact groups or infinite-dimensional representations.

For example on how to construct the unit morphism.

Are there good references for this topic?

PS: As a physicist I was especially looking for the case of the Poincaré group.