  If $\mathcal{C}$ is a cocomplete monoidal category in which tensoring from
both sides preserves coequalizers, then the category of monoids over
$\mathcal{C}$ is cocomplete. The same holds if $\mathcal{C}$ has regular
factorizations and tensoring only preserves regular epimorphisms. As an
application a lifting theorem for an adjunction with a monoidal right adjoint
to an adjunction between the respective categories of monoids is proved.
