definition. epic [kostecki2011introduction, 2.2] [tt-000C]

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An arrow \(f : X \to Y\) is epic if the diagram commutes, i.e. \(f \mathbin {\bullet } g_1 = f \mathbin {\bullet } g_2 \implies g_1 = g_2\), denoted \(f : X \twoheadrightarrow Y\).

"Epic" is short for "epimorphism", which is a generalization of the concept of surjective (onto) functions between sets.