  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.
