Consider the class $ob$ consisting of two distinct objects $x$ and $y$, and the class $mor$ of morphisms consisting of the ordered pairs $(x,x)$, $(y,y)$ and $(x,y)$. I would like to show that these classes constitute a category. I am stuck on what to do for a composition operator.
The composition operation is part of the definition of a category, so it doesn't make sense to ask whether these classes constitute a category; that will depend on how you define composition.
So I'll take the question to mean: How can we define composition for these morphisms such that the objects, the morphisms and the composition together will constitute a category?
We need identities, and (assuming that your notation is meant to indicate the source and target of the morphisms) there is only one candidate for each identity, so the compositional behaviour of $(x,x)$ and $(y,y)$ is determined by that. The only composition that doesn't include one of those would be $(x,y)\circ(x,y)$, and that doesn't occur because the source and target don't match.