As established in the last section, a matrix category consists of relations and inferences in a logic. Now, a matrix functor is a mapping of these relations and inferences, visualized from inner to outer.

Because a matrix category is a pseudo-bimodule, a matrix functor preserves composition actions only up to coherent isomorphism.

Let $[\![\mathbb{A}]\!]:\mathbb{A_0\to A_1}$ and $[\![\mathbb{B}]\!]:\mathbb{B_0\to B_1}$ be functors.

Let $\mathcal{R}_0:\mathbb{A}_0\,\|\,\mathbb{B}_0$ and $\mathcal{R}_1:\mathbb{A}_1\,\|\,\mathbb{B}_1$ be matrix categories.

A matrix functor $[\![\mathcal{R}]\!]:\mathcal{R_0\to R_1}$ is a span functor which forms a morphism of pseudobimodules in $\mathrm{Span\mathbb{C}at}$.

https://q.uiver.app/#q=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&embed

with invertible span transformations: the left join and right join

left join

$[\![\odot_\mathbb{A}]\!]:[\![\overline{\mathrm{a}}]\!]\odot_1 [\![R]\!]\cong [\![\overline{\mathrm{a}}\odot_0 R]\!]$

right join

$[\![\odot_\mathbb{B}]\!]:[\![R]\!]\odot_1 [\![\overline{\mathrm{b}}]\!]\cong [\![R\odot_0 \overline{\mathrm{b}}]\!]$

which together are natural with respect to the center associator:

and each is natural with respect to its own associator:

and each is natural with respect to its own unitor.

https://q.uiver.app/#q=WzAsNCxbMiwwLCJbXFwhW1JdXFwhXSJdLFswLDIsIltcXCFbXFxtYXRocm17aWQuQX1cXG9kb3QgUl1cXCFdIl0sWzAsMCwiXFxtYXRocm17aWR9LltcXCFbXFxtYXRocm17QX1dXFwhXVxcb2RvdCBbXFwhW1JdXFwhXSJdLFsyLDIsIltcXCFbUl1cXCFdIl0sWzAsMiwiXFx1cHNpbG9uX3tcXG1hdGhiYntBfX1eMSIsMV0sWzIsMSwiW1xcIVtcXG9kb3RfXFxtYXRoYmJ7QX1dXFwhXSIsMV0sWzMsMSwiXFx1cHNpbG9uX3tcXG1hdGhiYntBfX1eMCIsMV0sWzAsMywiIiwyLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=&embed

https://q.uiver.app/#q=WzAsNCxbMCwwLCJbXFwhW1JdXFwhXSJdLFsyLDIsIltcXCFbXFxtYXRocm17aWQuQn1cXG9kb3QgUl1cXCFdIl0sWzIsMCwiXFxtYXRocm17aWR9LltcXCFbXFxtYXRocm17Qn1dXFwhXVxcb2RvdCBbXFwhW1JdXFwhXSJdLFswLDIsIltcXCFbUl1cXCFdIl0sWzAsMiwiXFx1cHNpbG9uX3tcXG1hdGhiYntCfX1eMSIsMV0sWzIsMSwiW1xcIVtcXG9kb3RfXFxtYXRoYmJ7Qn1dXFwhXSIsMV0sWzMsMSwiXFx1cHNpbG9uX3tcXG1hdGhiYntCfX1eMCIsMV0sWzAsMywiIiwyLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=&embed

We summarize the concept of matrix functor.