CatDat

Implication Details

Assumptions: initial object, multi-complete

Conclusions: complete

Proof: Let $\C$ be a category with an initial object, and let $D : \I \to \C$ be a small diagram in $\C$. Since $\C$ has an initial object, the category $\Cone(D)$ of cones over $D$ also has an initial object. In particular, $\Cone(D)$ is connected, hence a multi-terminal object in it automatically becomes a terminal object. In other words, a multi-limit of $D$ is automatically a limit of $D$.