We illustrated the process of algebraic modeling by raising the question What are the laws of functional composition? We observed that the set of all functions $f\colon A\to A$ (i) satisfies the associative law and (ii) contains the identity function $\mathrm{id}_A\colon A\to A$ for which we have $\mathrm{id}_A\circ f = f = f\circ \mathrm{id}_A$. We asked whether these laws capture all the laws of functional composition. (Law = universally quantified equation.) To give an affirmative answer we created algebra models (=monoids) defined by the associative law and the unit laws. It is obvious that that the set of all functions $f\colon A\to A$ under composition and identity is a monoid (since we discovered these laws by examining concrete algebras of functions). This `obvious' fact can be stated: every concrete algebra of functions under composition and identity is an example of the abstract concept of a monoid. To show that there are no hidden laws of functional composition we proved a partial converse called The Cayley Representation Theorem (CRT) for monoids. It states: Every abstract monoid is embeddable in a concrete monoid of functions. The CRT implies that every universally quantified statement about functional composition is a consequence of the associative law and the unit laws. We pointed out that there are existentially quantified statements about functional composition that are not consequences of the associative law and the unit laws.

1. Functions are used to compare objects.
2. Homomorphisms between algebraic structures of the same type are functions between underlying sets that preserve the operations.

1. image
2. coimage
3. preimage, fiber, partition
4. inclusion map
5. natural map
6. induced map
7. injective/surjective/bijective function (= 1-1/onto/1-1 and onto)