Maybe you could construct an explicit exact sequence which describes something irl about which your friend has already an intuition?
Start with a tiny category (1,2,3 elements)
which is theoretically simple, and by a sequence of algebra-respecting surjections how a complex model of the real-world thing can be broken down (at different levels) into smaller bits with an easier logic to them. (Alternatively you could branch out in different directions and show how chain 1 captures this aspect of the model, whilst chain 2 captures that aspect of the model.) I don't have a ready example for that right now but I'll edit this answer if I think of one.
You could also spend time talking about "congruence"—just like it would be idiotic—pedantic beyond the worst human pedant—to treat two different hand-writings of the letter a as having different meanings, we want to be able to use the word "the same" in mathematics as we use it in normal language—like, "OK, not the same same, but, you know, basically the same". (And this needs some precise definition in order to give us, in mathematics, the freedom-of-movement we get gratis in eg English.) I like the familiar toddlers' play blocks as a metaphor for "The same like how?".
I'm not sure if this applies to your friend, but the simplest functor I can think of is between $(o,e,mathbb{Z}_+) longleftrightarrow (pos,neg,mathbb{R}_{times})$. Anyone who remembers grade-school arithmetic can follow the technical aspects of that relationship. (And you cover "the arrows changing along with the objects"—the relationship wouldn't stay the same if I substituted positives for evens without changing + to ×—and I think this is intuitively clear to anyone whose brain has been sufficiently jogged.)
I'm not sure if that captures what took you out of algebraic doldrums, but that's how I would explain category without busting out a Rube Goldberg of definitions first.
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