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Each capital letter is a type and each lower case letter is a function from one type to another. You can trace a path in the diagram to talk about a bunch of function calls (e.g following f then g then n represents n(g(f(a)))). This is called a diagram. Then the statement that a diagram is commutative says that if you trace any two paths that share a start and end, they are equal. So n(g(f(•))), s(r(l(•))), s(m(f(•))) are all paths/function calls you could make to get from A to C'. Since the diagram is said to be commutative, those paths are all equal. Being a monomoprishm, epimorphism, or isomorphism are all important properties of functions that say you're allowed to "cancel" on both sides of an equality. e.g. in general, if f(g(x))=f(h(x)), you can't conclude that g(x)=h(x). If f can be cancelled in that way, it's called a monomorphism. Similarly if g(f(x))=h(f(x)) lets you cancel the f to get g(x)=h(x), f is called an epimorphism. An isomorphism is both. This kind of thing let's you "walk backwards" along some paths in the diagram in certain situations. One flavor of theorem you might see in category theory (like the example five lemma[0]) looks like "look at this diagram. if g is an epimorphism and h is a monomorphism then f is an isomorphism". So if I know I can cancel this way and that way, I learn I can cancel this other way. [0] https://en.wikipedia.org/wiki/Five_lemma > The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism. |