By Fokkinga, M.M.; Jeuring, J.T.; Fokkinga, Maarten M

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Here are some more corollaries. Again we list them here only to show the importance of the concept. 43, due to the higher level of abstraction. 1. Adjoint functors determine each other “up to isomorphism”. We shall explain the concept of isomorphism later. More precisely, if A, B, G can be completed to an adjunction A, B, F, G, η, ε , then F is unique up to isomorphism. As a consequence, the existence of some F , η , ε for which the sextuple Set , Mon, F , G, η , ε (with G as in the above example) forms an adjunction, is equivalent to the existence of a monoid operation + + A (= F A) that has the categorical properties of ‘the monoid operation of sequences’.

These five laws become much more interesting in case the category is built upon another one, Set for example, and the typing is expressed as one or more equations in the underlying category Set . In particular the importance of law Fusion cannot be over-emphasised; we shall use it quite often. Exercise: give a fully calculational proof of init-Uniq, starting with the obligation ‘ f = g ’ at the top line of your calculation. Exercise: give a calculational proof of the equality ([1]) = (0) . Exercise: dualise the init-laws to final-laws; prove final-Fusion yourself, and see whether your proof is the dual of the one given above for init-Fusion.

X ✲ • •C • A cocone for D is: a family γA : DA → C of morphisms (for some C ), one for each A in D , satisfying: Df ; γB = γA for each f : A →D B . This condition is called ‘commutativity of the triangles’. Using naturality and constant functors there is a technically simpler definition of a cocone. Define C to be the constant functor, C x = C for each object x , and C f = id C for each morphism f . Now, each cocone for D is a natural transformation γ: D → .