\\(\textrm{dot} : M \times M \rightarrow M\\) is a homomorphism

\\(\iff \textrm{dot}((a, b)\cdot(c, d)) = \textrm{dot}(a, b)\cdot\textrm{dot}(c, d)\\)

\\(\iff \textrm{dot}(a\cdot c, b\cdot d) = \textrm{dot}(a, b)\cdot\textrm{dot}(c, d)\\)

\\(\iff (a\cdot c)\cdot (b\cdot d) = (a\cdot b)\cdot (c\cdot d)\\)

Set \\(a = d = 1\\), then \\(c\cdot d = (1\cdot c)\cdot (d\cdot 1) = (1\cdot d)\cdot (c\cdot 1) = d\cdot c\\)

OK I'm now thinking – a monoid is a category with one object, and a monoid homomorphism is a functor between monoids-as-categories. So can we generalise this to say something about categories-with-perhaps-many-objects when composition is functorial in some sense?

\\(\iff \textrm{dot}((a, b)\cdot(c, d)) = \textrm{dot}(a, b)\cdot\textrm{dot}(c, d)\\)

\\(\iff \textrm{dot}(a\cdot c, b\cdot d) = \textrm{dot}(a, b)\cdot\textrm{dot}(c, d)\\)

\\(\iff (a\cdot c)\cdot (b\cdot d) = (a\cdot b)\cdot (c\cdot d)\\)

Set \\(a = d = 1\\), then \\(c\cdot d = (1\cdot c)\cdot (d\cdot 1) = (1\cdot d)\cdot (c\cdot 1) = d\cdot c\\)

OK I'm now thinking – a monoid is a category with one object, and a monoid homomorphism is a functor between monoids-as-categories. So can we generalise this to say something about categories-with-perhaps-many-objects when composition is functorial in some sense?