Category theory and structural equality

Example 1.1.3 (Category theory). A third example comes from category theory:

• For two objects \(X\) and \(Y\) in a 1-category \(C\), the natural notion of equality between them is often not the strict set-theoretic equality, but rather the existence of an isomorphism \(f\colon X \iso Y\). Likewise, universal constructions in category theory are typically unique only ‘up to unique isomorphism’ rather than strictly equal.
• This pattern extends to functors: we wish to think of two functors \(F,G\colon C \to D\) as ‘the same’ whenever we are given a natural isomorphism \(\eta \colon F \iso G\). If we think of the individual isomorphisms \(\eta _X \colon F(X) \iso G(X)\) as ‘equalities’, then the naturality condition becomes a statement about structural properties of these equalities.
• In monoidal categories, the associativity and unitality conditions are expressed not as strict equalities but as natural isomorphisms: for objects \(X\), \(Y\), and \(Z\), we have an associator \(\alpha _{X,Y,Z}\colon (X \otimes Y) \otimes Z \iso X \otimes (Y \otimes Z)\) and unitors \(\lambda _X\colon 1 \otimes X \iso X\) and \(\rho _X\colon X \otimes 1 \iso X\). The coherence conditions (like the Mac Lane pentagon) then govern how these isomorphisms relate to each other.