Two categories sharing the same objects and morphisms Is there a natural example of two categories $\mathcal{C}$, $\mathcal{C}'$ which have the same class of objects and the same class of morphisms, including source and target maps, but different composition rules?
Of course it is easy to cook up an example, for example consider any set $X$ with two different monoid structures, this will produce two categories with one object with the desired properties. But this is not what I am looking for.
I would prefer an example where both categories $\mathcal{C}$ and $\mathcal{C}'$ are actually used in practice.
Background. While writing some basic stuff on categories, I have realized that often we only define categories by listing their objects and morphisms, saying almost nothing about the composition. In most cases this doesn't cause any confusion, because there is a "unique" reasonable way of composing the morphisms, but in general it may cause problems.

Consider categories whose objects are finite sets $X$ and whose morphisms $X \to Y$ are subsets of $X \times Y$. I can think of at least two interesting composition operations:


*

*Think of subsets of $X \times Y$ as $|X| \times |Y|$ matrices over the truth semiring, and perform matrix multiplication.

*Think of subsets of $X \times Y$ as $|X| \times |Y|$ matrices over $\mathbb{F}_2$, and perform matrix multiplication.


The first composition operation gives the category of finite sets and relations, while the second composition operation gives the category of finite-dimensional $\mathbb{F}_2$-vector spaces. 
