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Let ${Cat}^{\rightarrow\leftarrow} $ denote the category of all functors from $$\bullet \longrightarrow\bullet \longleftarrow \bullet$$ to $Cat$. The comma construction gives a functor $\left(.\downarrow .\right): Cat^{\rightarrow\leftarrow} \longrightarrow Cat $. The dual notion of this construction must be some functor which construct a category out of every functor from $\bullet \longleftarrow \bullet \longrightarrow\bullet$ to $Cat$. My first question is how this dual construction, preseunably called co-comma category is constructed. My second question is what are some example application of co-comma categories, parallel to use of comma categories in defining limits and colimits. Thanks.

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The comma category is a special kind of (strict) 2-limit: more precisely, it is the 2-limit of a diagram of shape $\bullet \rightarrow \bullet \leftarrow \bullet$, where the two outer vertices have weight $\mathbb{1}$ and the inner vertex has weight $\mathbb{2}$. Thus, it has the following universal property: given functors $F : \mathcal{C} \to \mathcal{E}$, $G : \mathcal{D} \to \mathcal{E}$, $H : \mathcal{A} \to \mathcal{D}$, $K : \mathcal{A} \to \mathcal{C}$ and a natural transformation $\phi : F K \Rightarrow G H$, there is a unique functor $\mathcal{A} \to (F \downarrow G)$ making the obvious diagrams commute.

The dual notion, then, is a special kind of (strict) 2-colimit, with a universal property of the following form: given functors $F : \mathcal{E} \to \mathcal{C}$, $G : \mathcal{E} \to \mathcal{D}$, $H : \mathcal{D} \to \mathcal{A}$, $K : \mathcal{C} \to \mathcal{A}$ and a natural transformation $\phi : K F \Rightarrow H G$, there is a unique functor $(F \star G) \to \mathcal{A}$ making the obvious diagrams commute. It is not clear to me what $(F \star G)$ looks like in general, but at least in some simple cases, it is the disjoint union of $\mathcal{C}$ and $\mathcal{D}$ together with a new arrow joining the object $F E$ in $\mathcal{C}$ to the object $G E$ in $\mathcal{D}$, these making various diagrams commute. For example, if $\mathcal{D} = \mathbb{1}$, then the co-comma object $(F \star G)$ is the result of freely adjoining to $\mathcal{C}$ a cocone from $F$ to a new object.

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