Just as a matrix profunctor is a bimodule of weave profunctors, a matrix transformation is a homomorphism of these bimodules, which coheres with the joins of the source and target matrix functors.

Let $\mathbb{[\![X]\!], [\![Y]\!], [\![A]\!],[\![B]\!]}$ be functors, $f_0:\mathbb{X_0}\,|\,\mathbb{A_0}$, $f_1:\mathbb{X_1}\,|\,\mathbb{A_1}$, $g_0: \mathbb{Y}_0\,|\, \mathbb{B}_0$, $g_1: \mathbb{Y}_1\,|\, \mathbb{B}_1$ profunctors, and $[\![f]\!]:f_0\Rightarrow f_1$, $[\![g]\!]:g_0\Rightarrow g_1$ transformations.

Let $\mathcal{Q}_0(\mathbb{X_0,Y_0})$, $\mathcal{Q}_1(\mathbb{X_1,Y_1})$, $\mathcal{R}_0(\mathbb{A_0,B_0})$, $\mathcal{R}_1(\mathbb{A_1,B_1})$ be matrix categories, $[\![\mathcal{Q}]\!], [\![\mathcal{R}]\!]$ matrix functors, and $i_0(f_0,g_0)$, $i_1(f_1,g_1)$ matrix profunctors.

A matrix transformation $[\![i]\!]:i_0\Rightarrow i_1$ is a span transformation

which coheres with the left and right joins of $[\![\mathcal{Q}]\!]$ and $[\![\mathcal{R}]\!]$.

We summarize the concept of matrix transformation.