See here (especially this draft article) for definition of cross-composition product and quasi-cartesian functions.

**Conjecture 1** * Cross-composition product (for small indexed families of relations) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation $latex {\mathfrak{S}_0}&fg=000000$ of binary relations to the quasi-cartesian situation $latex {\mathfrak{S}_1}&fg=000000$ of pointfree funcoids over posets with least elements. *

**Conjecture 2** * Cross-composition product (for small indexed families of pointfree funcoid between posets with least elements) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation $latex {\mathfrak{S}_0}&fg=000000$ of pointfree funcoids over posets with least elements to the quasi-cartesian situation $latex {\mathfrak{S}_1}&fg=000000$ of pointfree funcoids over posets with least elements. *

**Conjecture 3** * Cross-composition product (for small indexed families of reloids) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation $latex {\mathfrak{S}_0}&fg=000000$ of reloids to the quasi-cartesian situation $latex {\mathfrak{S}_1}&fg=000000$ of pointfree funcoids over posets with least elements. *

**Remark 1** * The three above conjectures are unsolved even for product of two multipliers. *

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