If C is a monoidal category with reflexive coequalizers which are preserved by tensoring from both sides, then the category MonC of monoids over C has all coequalizers and these are regular epimorphisms in C. This implies that MonC has all colimits which exist in C, provided that C in addition has (regular epi, jointly monomorphic)-factorizations of discrete cones and admits arbitrary free monoids. A further application is a lifting theorem for adjunctions with a monoidal right adjoint whose left adjoint is not necessarily strong to adjunctions between the respective categories of monoids.
Keywords: Monoids in monoidal categories, (reflexive) coequalizers, (regularly) monadic functors, monoidal functors
2010 MSC: Primary 18D10, Secondary 18A30
Theory and Applications of Categories, Vol. 34, 2019, No. 17, pp 456-467.