Implication Details

Assumptions: finite coproducts, multi-complete

Conclusions: complete

Reason: Let $D\colon \mathcal{S} \to \mathcal{C}$ be a small diagram in a category $\mathcal{C}$ . Since $\mathcal{C}$ has finite coproducts, the category $\mathbf{Cone}(D)$ of cones over $D$ has finite coproducts. In particular, $\mathbf{Cone}(D)$ is connected, hence a multi-terminal object in it automatically becomes a terminal object.