  Finite depth quantum circuits provide an equivalence relation between gapped
phases. Moreover, there can be nontrivial domain walls either within the same
gapped phase or between different gapped phases, whose equivalence relations
are given by finite depth quantum circuits in one lower dimension. In this
paper, we use such unitary equivalence relations to study the fusion of
one-dimensional gapped phases. In particular, we use finite depth circuits to
fuse two gapped phases, local unitaries to fuse two domain walls, and a
combination of both to fuse gapped phases with domain walls. This provides a
concrete illustration of some simple aspects of the `higher-category' structure
of gapped defects in a higher-dimensional trivial gapped bulk state.
