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Example We define a category $\operatorname{Set}_{\ast }$ as follows:

  • The objects of $\operatorname{Set}_{\ast }$ are pairs $(X,x)$, where $X$ is a set and $x \in X$ is an element.

  • A morphism from $(X,x)$ to $(Y,y)$ in $\operatorname{Set}_{\ast }$ is a function $f: X \rightarrow Y$ satisfying $f(x) = y$.

We will refer to $\operatorname{Set}_{\ast }$ as the category of pointed sets. The construction $(X,x) \mapsto X$ determines a left covering functor $\operatorname{Set}_{\ast } \rightarrow \operatorname{Set}$ (for a more general assertion, see Remark