Bihom hopf algebras viewed as hopf monoids

We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them Lax+Oplax0-monoidal. Dually, we consider Lax0Oplax+-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define Lax+0Oplax0+-duoidal categories with compatible Lax+Oplax0- and Lax0Oplax+-monoidal structures. We introduce comonoids in Lax+Oplax0-monoidal categories, monoids in Lax0Oplax+-monoidal categories and bimonoids in Lax+0Oplax0+-duoidal categories. Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a Lax+0Oplax0+-duoidal category D to any symmetric monoidal category V. The unital BiHom-monoids, counital BiHom-comonoids, and unital and counital BiHom-bimonoids in V, due to Grazianu et al. are identified with the monoids, comonoids and bimonoids in D, respectively.SCOPUS: cp.p