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Construction (Slice Categories over Objects). Let $\operatorname{\mathcal{C}}$ be a category containing an object $S$. We define a category $\operatorname{\mathcal{C}}_{/S}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}_{/S}$ are pairs $(X, f)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $f: X \rightarrow S$ is a morphism in $\operatorname{\mathcal{C}}$.

  • If $(X,f)$ and $(Y, g)$ are objects of $\operatorname{\mathcal{C}}_{/S}$, then a morphism from $(X,f)$ to $(Y,g)$ in the category $\operatorname{\mathcal{C}}_{/S}$ is a morphism $u: X \rightarrow Y$ in the category $\operatorname{\mathcal{C}}$ satisfying $f = g \circ u$. In other words, morphisms from $(X,f)$ to $(Y,g)$ are given by commutative diagrams

    \[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{u} \ar [dr]_-{f} & & Y \ar [dl]^-{g} \\ & S & } \]

    in the category $\operatorname{\mathcal{C}}$.

  • Composition of morphisms in the category $\operatorname{\mathcal{C}}_{/S}$ is given by composition of morphisms in the category $\operatorname{\mathcal{C}}$.

We will refer to $\operatorname{\mathcal{C}}_{/S}$ as the slice category of $\operatorname{\mathcal{C}}$ over $S$.