  This paper studies questions of coherence and strictification related to
self-similarity - the identity $S\cong S\otimes S$ in a (semi-)monoidal
category. Based on Saavedra's theory of units, we first demonstrate that strict
self-similarity cannot simultaneously occur with strict associativity -- i.e.
no monoid may have a strictly associative (semi-)monoidal tensor, although many
monoids have a semi-monoidal tensor associative up to isomorphism. We then give
a simple coherence result for the arrows exhibiting self-similarity and use
this to describe a `strictification procedure' that gives a semi-monoidal
equivalence of categories relating strict and non-strict self-similarity, and
hence monoid analogues of many categorical properties. Using this, we
characterise a large class of diagrams (built from the canonical isomorphisms
for the relevant tensors, together with the isomorphisms exhibiting the
self-similarity) that are guaranteed to commute.
