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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 ...


2

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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