# Do morphisms need to make full use of their domain (category theory)?

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!

• That's what I thought too, but being inexperienced and self-studying I wanted to make sure I understood the basic basics. One last question: is the $H:\Sigma\to\Sigma'$ notation inappropiate here? I.e. should I write $\Sigma\stackrel{H}{\to}\Sigma'$ and $H:A\to A'$ instead? How do I avoid ambiguity / confusion? – BlondCafé Dec 13 '19 at 16:10
• Well $H$ is a morphism in the category, which corresponds to a map of sets, which you could denote by $H$ as well if it's clear (but it's an abuse of notation), or you could find a second notation – Maxime Ramzi Dec 13 '19 at 16:19