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Variant (Coslice Categories under 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 $f: S \rightarrow X$ 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 $g = f \circ u$. In other words, morphisms from $(X,f)$ to $(Y,g)$ are given by commutative diagrams

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

    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 coslice category of $\operatorname{\mathcal{C}}$ under $S$.