definition. subobject, Sub [kostecki2011introduction, 2.10] [tt-004J]

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A subobject of any object is defined as an equivalence class of monic arrows into it.

The class of subobjects of an object \(X\) is denoted as \[ \operatorname {Sub}(X):=\{[f] \mid \operatorname {cod}(f)=X \wedge f \text { is monic }\}. \]