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Suppose I am working with a category of objects such that each object $X$ in the category has as a trivial automorphism group. An example of such an object would be genus zero curves with $n$-punctures, i.e., $(\mathbb{P}_k^1, (s_i)_{i=1}^n \in k )$.
Let $F(S)/{\sim}$ represent families of thes...