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Every category $\mathcal K$ has a free completion $\mathcal P \mathcal K$under colimits and a free completion $\Sigma\mathcal K$ under coproducts. Anumber of properties of $\mathcal K$ transfer to $\mathcal P \mathcal K$ and$\Sigma\mathcal K$ (e.g., completeness or cartesian closedness). We prove that$\mathcal P\mathcal K$ is always a pretopos, but, for $\mathcal K$ large,seldom a topos. Moreover, for complete categories $\mathcal K$ we prove that$\mathcal P\mathcal K$ is locally cartesian closed whenever $\mathcal K$ isadditive or cartesian closed or dual to an extensive category. We also studythe question whether $\mathcal P \mathcal K$ is (co)wellpowered. The answer isaffirmative for "set-like" categories. But for a number of categories $\mathcalK$ the answer turns out to be negative.

How nice are free completions of categories?