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The grammar is "a category is tensored over a monoidal category"; this is a generalization of a set being equipped with an action of a monoid, or an abelian group being equipped with an action of a ring. In full generality you should provide the tensoring, but sometimes if you require enough it exists uniquely. The general pattern of the uniqueness results ...


Here is how it works in general: If $\mathcal{C}$ is a category and $P$ is an object (in your case, a final object), then the forgetful functor $P/\mathcal{C} \to \mathcal{C}$ creates limits. In particular, if $\mathcal{C}$ is complete, then $P/\mathcal{C}$ is complete, too. To see this, consider a diagram $(P \to X_i)$ in $P/\mathcal{C}$ and consider a ...

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