I'm new to category theory, and have a (I assume very basic) question that's probably best explained with an example.
Say we have a category $C$ with pairs of sets as objects, e.g. $$\Sigma = (A,B) = (\{a_0,a_1,\ldots\},\{b_0,b_1,\ldots\})$$ and $$\Sigma' = (A',B') = (\{a '_0,a'_1,\ldots\},\{b'_0,b'_1,\ldots\}).$$ Now, is it okay if I say the morphisms of $C$ are functions (i.e. set-theoretic functions) between the first "component" of the objects? E.g. $H:\Sigma\to\Sigma'$, $H(a_0)=a'_0$, $H(b_0)=\text{undefined}$, $H(A)\subseteq A'$, $H(B)=\varnothing$, etc.
Any help would be much appreciated!
Yes, why wouldn't it be ?
Just define composition and check the axioms for a category, and you'll see that it is one (although in your situation it will be equivalent to the category of sets)