Implication Details

Claim: If a category is total, then it is cocomplete.

Proof: If the category $\C$ is locally small and total, then the Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ makes $\C^{\op}$ into a reflective subcategory of the presheaf category $[\C, \Set]$, where the latter is complete. For a general total category $\C$, use the equivalence to a locally small, total category.