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Lenses are a category theoretic construct and are used in a wide variety ofapplications. Symmetric lenses compose to, of course, form new symmetriclenses. Symmetric lenses are usually represented as spans of asymmetric lenses.In many applications, the fact that a symmetric lens might also be representedas a cospan of asymmetric lenses is important, especially for implementationpurposes. However, the composition of symmetric lenses does not preserve theproperty that the lenses can be represented by cospans -- two such symmetriclenses may (and frequently do) compose to form a symmetric lens which cannot berepresented as a cospan of asymmetric lenses. Thus preserving the factorisationto show how cospans of asymmetric lenses might be used in implementationsbecomes important. In 2018, the first work on multilenses was begun.Multilenses can be represented as multi-spans of asymmetric lenses (oftencalled 'wide spans', these are spans with an arbitrary finite number of legs).In this paper we analyse a small but realistic example of a supply chain inwhich the cospan representations would be 'composed away' by ordinary symmetriclens composition, and introduce a new kind of composition which we call'fusion' in which two ordinary symmetric lenses (spans with two legs) fuse toform a multilens with three legs preserving the cospan representations, andmore generally, two symmetric multilenses, spans with say m and n legs, fuse toform a symmetric multilens with m+n-1 legs, again preserving cospanrepresentations.

The more legs the merrier: A new composition for symmetric (multi-)lenses