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 $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal. Dually, we consider $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal categories with compatible $\mathsf{Lax}^+\mathsf{Oplax}^0$- and $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal structures. We introduce comonoids in $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal categories, monoids in $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories and bimonoids in $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$- duoidal categories. Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal category $\mathsf D$ to any symmetric monoidal category $\mathsf V$. The unital $\mathsf{BiHom}$-monoids, counital $\mathsf{BiHom}$-comonoids, and unital and counital $\mathsf{BiHom}$-bimonoids in $\mathsf V$ are identified with the monoids, comonoids and bimonoids in $\mathsf D$, respectively.

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