  The goal of this paper is to prove coherence results with respect to
relational graphs for monoidal endofunctors, i.e. endofunctors of a monoidal
category that preserve the monoidal structure up to a natural transformation
that need not be an isomorphism. These results are proved first in the absence
of symmetry in the monoidal structure, and then with this symmetry. In the
later parts of the paper the coherence results are extended to monoidal
endofunctors in monoidal categories that have diagonal or codiagonal natural
transformations, or where the monoidal structure is given by finite products or
coproducts. Monoidal endofunctors are interesting because they stand behind
monoidal monads and comonads, for which coherence will be proved in a sequel to
this paper.
