definition. initial, terminal and null objects [kostecki2011introduction, 2.6] [tt-000I]

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An initial object in a category \({\cal C}\) is an object \(\mathrm {0} \in \operatorname {Ob}({\cal C})\) such that for any object \(X\) in \({\cal C}\), there exists a unique arrow \(\mathrm {0} \to X\). It's also called a universal object, or a free object.

A terminal object in a category C is an object \(\mathrm {1} \in \operatorname {Ob}({\cal C})\) such that for any object \(X\) in \({\cal C}\), there exists a unique arrow \(X \to \mathrm {1}\). It's also called a final object, or a bound object.

Diagramatically,

A null object is an object which is both terminal and initial, confusingly, it's also called a zero object.