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Let $\omega:\mathcal{C} \to \mathrm{Set}^I$ be a topological functor where $I$ is a set (i.e. a discrete small category). I have to consider maps $f:X\to Y$ of $\mathcal{C}$ such that $X$ is equipped with the $\omega$-initial structure and such that, with $\omega(f)=(f_i)$, each $f_i$ is surjective.

Does it have a name ?

I want to call them initially epic maps. Of course I browsed the book Abstract and concrete categories: The joy of cats.

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