I'm moving my first steps in CT and suddenly after reading about functors, this question came up in my mind:

Let $F \colon A \to B$ a functor between two categories $A$ and $B$, is it true that for any arrow $f \colon a \to a'$ in $A$, 

![enter image description here](https://i.stack.imgur.com/UYBXU.png)

is commutative?

According to me the definition of functor doesn't involve nor implies this identity. Nevertheless, due to my restricted experience in CT, I cannot find a counterexample. 

I hope the clarification will help me build a strong "set" of example/counterexample in this field.

Thanks in advance.