Exercise 7.6

To answer 1. let’s start with the fact that \(f\) is a monomorphism i.e. a left-cancellative morphism:

\[ \begin{align} f∘a = f∘b & ⇒ a = b \end{align} \]

In Set we know that all of \(f,a,b\) are simply functions, so we can re-express composition as:

\[ \begin{align} f(a()) = f(b()) & ⇒ a = b \end{align} \]

Notice we make \(Z=X^0\) or the empty tuple so that we can see \(a\) and \(b\) as nullary operations (language from Pointed set). To make these maps from e.g. \(Z=\underline{2}\) would only add unnecessary complexity i.e. indirection, essentially precomposing to produce something like:

\[\begin{split} \begin{align} f∘a∘z & = f∘b∘z \\ f(a(z)) & = f(b(z)) \end{align} \end{split}\]

Instead, let’s simply see \(a,b\) as objects again so that:

\[ \begin{align} f(a) = f(b) & ⇒ a = b \end{align} \]

See Injective function for the formal requirements to be an injective function, in particular Injective function § Proving that functions are injective. This is the definition of an injective function.

To answer 2. we can simply follow the same steps in reverse.