Let $\mathcal{R}_0: \mathbb{A}_0\,\|\, \mathbb{B}_0$ and $\mathcal{R}_1: \mathbb{A}_1\,\|\, \mathbb{B}_1$ be span categories.
A span functor from $\mathcal{R}_0$ to $\mathcal{R}_1$ is
a pair of functors $[\![\mathbb{A}]\!]: \mathbb{A_0\to A_1}$ and $[\![\mathbb{B}]\!]: \mathbb{B_0\to B_1}$,
and a functor $[\![\mathcal{R}]\!]: \mathcal{R_0\to R_1}$ such that the two squares commute,
i.e. for any $R:\mathcal{R}_0$ over $\mathrm{A_0,B_0}$ we have that $[\![R]\!]:\mathcal{R_1}$ lies over $[\![\mathrm{A}]\!], [\![\mathrm{B}]\!]$.
Just as a span category is a matrix of categories, a span functor is a matrix of functors.
Let $\mathcal{R}_0:\mathbb{A_0\times B_0}\to \mathbb{C}\mathrm{at}$ and $\mathcal{R}_1:\mathbb{A_1\times B_1}\to \mathbb{C}\mathrm{at}$ be displayed categories, and $[\![\mathbb{A}]\!]: \mathbb{A_0\to A_1}$, $[\![\mathbb{B}]\!]: \mathbb{B_0\to B_1}$ be functors.
A displayed functor $[\![\mathcal{R}]\!]:\mathcal{R_0}\Rightarrow \mathcal{R_1}([\![\mathbb{A}]\!], [\![\mathbb{B}]\!])$
https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXG1hdGhiYntBXzBcXHRpbWVzIEJfMH0iXSxbMiwwLCJcXG1hdGhiYntBXzFcXHRpbWVzIEJfMX0iXSxbMCwyLCJcXG1hdGhybXtDYXR9Il0sWzIsMiwiXFxtYXRocm17Q2F0fSJdLFswLDIsIlxcbWF0aGNhbHtSXzB9IiwxXSxbMSwzLCJcXG1hdGhjYWx7Ul8xfSIsMV0sWzIsMywiXFxtYXRocm17RnVufSIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImJhcnJlZCJ9fX1dLFswLDEsIltcXCFbXFxtYXRoYmJ7QX1dXFwhXVxcdGltZXMgW1xcIVtcXG1hdGhiYntCfV1cXCFdIl0sWzcsNiwiW1xcIVtcXG1hdGhjYWx7Un1dXFwhXSIsMSx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH19XV0=&embed
gives for each pair:
| objects | $\mathrm{A_0}:\mathbb{A}_0, \mathrm{B_0}:\mathbb{B}_0$ | a functor | $\![\mathcal{R}]\!: \mathcal{R_0}(\mathrm{A_0,B_0})\to \mathcal{R_1}([\![\mathrm{A_0}]\!],[\![\mathrm{B_0}]\!])$ |
|---|---|---|---|
| morphisms | $\mathrm{a_0}:\vec{\mathbb{A}}_0, | ||
| \mathrm{b_0}:\vec{\mathbb{B}}_0,$ | a transformation | $\![\vec{\mathcal{R}}]\!: \vec{\mathcal{R}}_0(\mathrm{a_0,b_0})\Rightarrow \vec{\mathcal{R}}_1([\![\mathrm{a_0}]\!], [\![\mathrm{b_0}]\!])$ | |
| composable pairs | $\mathrm{(a_1,b_1), (a_2,b_2)}$ | an equality | $\![\vec{\mathcal{R}}]\!\cdot \![\vec{\mathcal{R}}]\! = \![\vec{\mathcal{R}}]\!$ |
| objects | $\mathrm{A_0,B_0}$ | an equality | $\![\vec{\mathcal{R}}]\! = \mathrm{id}.\mathcal{R}_1([\![\mathrm{A_0}]\!], [\![\mathrm{B_0}]\!])$ |
Proposition. Let $\mathbb{A}_0\leftarrow \mathcal{R_0}\to \mathbb{B_0}$ and $\mathbb{A}_1\leftarrow \mathcal{R_1}\to \mathbb{B_1}$ be span categories. Let $\![\mathcal{R}]\!:\mathcal{R_0\to R_1}$ be a span functor over $[\![\mathbb{A}]\!], [\![\mathbb{B}]\!]$. Inverse image along $[\![\mathcal{R}]\!]$ determines a displayed functor $[\![\mathcal{R}]\!]:\mathcal{R_0}\Rightarrow \mathcal{R_1}([\![\mathbb{A}]\!], [\![\mathbb{B}]\!])$.
We now expound this idea, in color syntax.
A functor is a transversal morphism in $\mathrm{Span\mathbb{C}at}$, drawn as a string with a small “bubble” pointer, filled with the color of its source. A span functor, like a transformation, is drawn as a solid black bead, to distinguish from the “open” bead of a span profunctor.