Functional equality

sona@stau.space (Sona Tau Estrada Rivera) — Tue, 11 Aug 2026 10:07:36 +0000

While learning about basic category theory a reocurring question kept coming up again and again:

How do I know whether two morphisms are equal?

This is the question this essay explores.

Let’s start somewhere familiar, $\mathbf{Set}$. We know the morphisms in this category very well, they are just functions. And we have a very good notion of equality of functions. Two functions $f : A \to B$ and $g : A \to B$ are equal when for all $x \in A$ we have that $f(x) = g(x)$. In other words, if when evaluated at every point of $A$ they are equal then we know that $f$ and $g$ are equal. We don’t want equality of functions though, we want to define equality of morphisms. So, we need to abstract this using categorical terms.

Yoneda on stau.space

Functional equality