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For categories $\mathbb{A}$ and $\mathbb{B}$ I know that the functor category $\mathbb{B}^{\mathbb{A}}$ is a good way to model the functors from $\mathbb{A}$ to $\mathbb{B}$ (because the objects of $\mathbb{B}^{\mathbb{A}}$ correspond to the functors from $\mathbb{A}$ to $\mathbb{B}$). My question is, what if I want to model all functors into the category $\mathbb{B}$ (functors with any source category) ? In other words, is there a nice category which has the collection of all functors into $\mathbb{B}$ as its collection of objects ?

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The slice category $\mathbf{CAT}/\mathbb B$ has as its objects pairs $(\mathbb A, A \colon \mathbb A \to \mathbb B)$ and as morphisms functors $F \colon \mathbb A \to \mathbb A'$ making the triangle commute (i.e. $F \circ A = A'$). One can also consider triangles that commute only up to natural isomorphism, which may be more appropriate in this case, since $\mathbf{CAT}$ is actually a 2-category. This is given by the (underlying 1-category) of the slice 2-category.

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  • $\begingroup$ Ah yes, that's it. It just came to me as well ! $\endgroup$ Jul 29 at 12:57

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