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The Kernel of a Relational Morphism of Categories

This paper introduces the notion of the kernel of a relational morphism of categories. This is the second step in the working out of a useful definition of the block product of two C-varieties which in turn is needed for the theory of decompositions of monoids by means of wreath or block products. In a previous work, the author defined the construction of the block product of two categories and the strong semidirect product functor. In this second part, we construct a left adjoint of the strong semidirect product functor which is the starting point of our construction of the kernel of a relational morphism. With these notions we define the block product of two C-varieties and prove a kernel theorem in the category settings which extends a work of J. Rhodes and B. Tilson.