Definition. Free category with finite products monad [mde-RMGV]

The free category with strictly associative finite products monad \(T : \mathbf {Cat} \to \mathbf {Cat}\) sends a category \(\mathbb {C}\) to the category \(T\mathbb {C}\) with

• \(\mathbb {C}_{\text {obj}}\) comprises lists of \(\mathbb {C}\) objects,
• \(T\mathbb {C}((x_1, ..., x_m),(y_1,...,y_n)) := \prod _{1 \leqslant i \leqslant n} \coprod _{1 \leqslant j \leqslant m} X(x_j, x_i).\)

Intuitively, lists of objects are formal products, and the maps between products are determined by choosing, for each index \(i\) of the output, a projection \(j\) of the input and a morphism \(x_j \to y_i\).

This extends to monads on \(\mathcal {V}\)-\(\mathbf {Cat}\) and \(\mathcal {V}\)-\(\mathbf {Prof}\).