Commutative diagrams and equality of composition

[Mr Davis 97]

I am a little bit confused on how commutative diagrams show equality of two morphisms. For example, one can imagine the diagram for hf = kg, where composing f and g is the same morphism as composing h and k:
https://upload.wikimedia.org/wikipedia/commons/9/91/Commutative_square.svg

Why does the commutativity of this diagram imply equality of the composition h and f, and k and g? Wouldn't the commutativity just show that hf and kg have the same domain and codomain but are not necessarily the same map?

 

[fresh_42]

The commutativity works elementwise: $h(f(a))=k(g(a))\,\forall\,a\in A$. There might happen different things in between (at $B$ and $C$), but the compositions are equal $h\circ f = k \circ g$.